Overview of Quantum Physics Concepts

Hello, welcome once again to my engineering physics channel. This is Sanju Badhe from K.J. Somaya Institute of Engineering and Information Technology. From this session, we are starting with discussion on quantum physics. In the first session, we will be getting introduced to De Broglie hypothesis and in further sessions, we will be studying Heisenberg uncertainty principle and Schrodinger's equations. This is the flow of discussion that we will be following for the discussion. First, we will get introduced to de Broglie hypothesis of matter waves. We will discuss properties of the matter waves. Then we will get introduced to the concept of wave packet Heisenberg and uncertainty principle, one important application of Heisenberg uncertainty principle which explains why electron cannot exist in the nucleus. Then we will derive Schrodinger's time dependent equation and Schrodinger's time independent equation and we will consider a case of a particle trapped in one dimensional infinite potential well. And at the end, we will be discussing application of quantum physics, one important application quantum coding. The concept of quantum physics was first used by Max Planck to explain the results obtained in black body radiation experiment. A black body is designed to absorb 99.9 percent of the radiation incident on it. Now, how this black body can be obtained? This was done with a hollow sphere with a narrow hole. Inner surface of this hollow sphere is coated with carbon black. When the radiation enters through a narrow hole, it hits the wall of the sphere and gets almost fully absorbed. Remaining radiation is reflected back and collide again with the wall. After multiple reflections, almost all the radiation is absorbed by this black body. Due to the absorption of radiation, temperature of the body increases and then it starts emitting the radiation from the narrow hole. This radiation contains all the wavelengths. This is called as the black body radiation. When intensity versus wavelength curve is plotted, it is found that radiation for higher wavelength is very low and for shorter wavelength the intensity is more. For a particular wavelength called lambda max at a given temperature, the intensity is maximum. The wavelength for which maximum energy is radiated is called as the peak wavelength. As you can see in the graph, all the wavelengths are emitted but there is maximum intensity only for the peak wavelength. The black body will appear of the color corresponding to that peak wavelength. If you increase the temperature, what happens to this peak wavelength? This peak wavelength is shifting to the left. So, for temperature equal to 3000 Kelvin, peak wavelength is somewhere around here. For 4000 Kelvin, the peak wavelength shifts to the left. left side. For 5000 Kelvin, again the peak wavelength shifts to the left side. For 6000, it further shifts to the left side. So, if we increase the temperature of the black body, the lambda max, the peak wavelength shift towards the left side and the color of the black body will change accordingly. This shift in the peak wavelength was explained by Vance displacement law which states that the peak wavelength is inversely proportional to the temperature. So as the temperature increases, the peak wavelength will shift towards the left side. As the temperature increases, the peak wavelength will reduce. So, this Vance displacement law could explain the shift in the peak wavelength as the temperature increases but this law could not explain the part of the curve towards the longer wavelength. So, for higher wavelengths, this law could not explain the behavior of the curve. Also, if the temperature is very close to 0, then the peak wavelength will become infinite. But when the temperature is lowered, the radiation curve changes the behavior which also could not be explained by this Vann's law. Rayleigh and Zinn tried to explain the blackbody radiation curve using their Rayleigh-Zinn's radiation law which is based on classical statistics. According to this law, the energy density or the intensity is directly proportional to square of the frequency of radiation. That means it is inversely proportional to square of the wavelength. This Rayleigh-Zinn's radiation law could explain the behavior of this radiation curve. radiation curve on the higher wavelength side. So, for higher wavelengths it is applicable, but when it comes to the lower wavelength, this law could not explain the behavior of the radiation curve. So, Vann's law could explain the part of the curve on the lower wavelength side and Rayleigh-Jeans radiation law could explain the part of this radiation curve, black body radiation curve on the higher wavelength side. But the complete curve could not be explained by any of these two laws. So, we have seen Vance displacement law could explain only the shorter wavelength side of this blackbody radiation curve and Rayleigh-Jeans law could explain only the longer wavelength side of this blackbody radiation curve. The entire curve could not be explained. explained by any of these two laws. To explain the black body radiation curve, Max Planck came forward and he suggested that radiated energy must be depending on frequency of radiation and he represented this energy radiated E to be equal to h nu where h is called as Planck's constant. Max Planck calculated the average energy of radiation using Maxwell Boltzmann distribution law and this energy was calculated as E average equal to h nu upon E raise to h nu by kT minus 1. Max Planck simply replaced the average energy kT in Rayleigh-Jeans formula by this new value calculated by him and the Rayleigh-Jeans formula now becomes rho V, the energy density density is equal to 8 pi nu square upon c cube as it is you can see into the average value calculated by max Planck, average energy value calculated by max Planck h nu upon e raise to h nu by kT minus 1. So, this will become rho v equal to 8 pi h nu cube upon c cube into bracket 1 upon e raise to h nu by kT minus 1. This is called as Planck's radiation law and this Planck's radiation law now gives correct explanation for the blackbody radiation curve. So, while explaining the black body radiation curve, Planck has introduced this revolutionary idea which says energy of the radiation depends on the frequency and is given by E is equal to h nu. Einstein used this idea to explain his photoelectric effect. This E is equal to h nu is called as quantum of light energy. Max Planck was awarded Nobel Prize in in 1918 for this discovery. Einstein used Planck's idea which says energy radiated must be depending on the frequency of radiation and E is equal to h nu. Einstein confirmed that light consists of discrete unit of energy which are known as photons and each photon has energy E is equal to h nu where nu is frequency of the radiation. This photoelectric effect confirmed the dual nature of light. So, light behaves as particle as well as wave. As far as macroscopic particles which are bigger in size, moving with comparatively lower speeds are considered, the motion of such particles can be explained by classical theory of mechanics. But when it comes to microscopic particles like electrons, protons, etc which move with very high speed, this classical theory of mechanics fails to explain the motion of such particle. So, it was needed to develop a new theory which could explain the motion of such microscopic particles. So, quantum mechanics was developed from quantum theory to explain the properties associated with such particles. De Broglie wave particle duality hypothesis, Heisenberg uncertainty principle and Schrodinger equation provide base on which the quantum mechanics is built. So, we will be discussing this De Broglie hypothesis, Heisenberg uncertainty principle and Schrodinger's equation one by one in the coming sessions. So, we have seen photoelectric effect introduced firm evidence of a particle nature of light. Data There are some phenomena like interference, diffraction, polarization which could be explained by considering only wave nature of light. But photoelectric effect, to explain the photoelectric effect, particle nature has to be considered. It shows the light exhibits a dual nature. Deeprogli just took the same idea. He said if dual nature exists for light, the dual nature must exist for all type of matter. Because the nature is symmetric, if one phenomenon is applicable to certain type of matter, then it is called as dual nature. matter, the same phenomenon should be applicable to all type of matter. And he stated his hypothesis as there is a wave associated with every moving particle moving with velocity v and the wavelength this wave is given by lambda is equal to h upon p that is h upon mv where m is mass of the particle, v is velocity with which the particle is moving and h is Planck's constant. So according to de Broglie hypothesis, every moving particle has a wave associated with it. This is the first part and wavelength of this wave is given by lambda is equal to h upon p, this is the second part of the hypothesis. Why it is called as hypothesis? Because the firm evidence was not available. It is just a proposed explanation made on the basis of some limited evidence as a starting point for further investigation. De Broglie just said if wave particle duality exist for light, then wave particle duality must exist for all type of particles. So, there should be a wave associated with electron also. There should be a wave associated with any other particle which is moving with certain velocity v and wavelength of that wave should be lambda equal to h upon p. So, this is what the Broglie hypothesis says. De Broglie relation can be supported by using Planck's theory and Einstein's theory. Energy of a photon according to Planck's theory is given by E is equal to h nu where h is Planck's constant and nu is frequency of radiation. And if we consider photon as a particle of mass m, then its energy as per Einstein's mass energy relation is given as E is equal to mc square. So, both h nu and mc square are energies of the photon and therefore we can equate them. h nu equal to mc square, let us call this as equation number 3. The photon is travelling with velocity c in free space. So, velocity of light is c. So, photon will travel with velocity equal to c. So, its momentum p is given by m into c. So, momentum of photon is equal to m into c. Let us call this as equation number 4. De Broglie supported his hypothesis by using Planck's relation E is equal to h nu and Einstein's mass energy relation E is equal to mc square. So h nu and mc square both are energies of the photon. So h nu can be equated with mc square. The momentum of the photon P is equal to mc mass into velocity. So, we can divide equation number 3 by equation number 4. So, h nu by p is equal to mc square upon mc. So, 1c will get cancelled here, mm will get cancelled with each other. So, h nu upon p is equal to c or h upon p is equal to c by nu and c by nu is nothing but lambda, the wavelength. So, lambda is equal to h upon p. So, de Broglie just said if E is equal to h nu that is Planck's relation is correct and E is equal to mc square that is Einstein's relation is correct then lambda must be equal to h upon p. So, this lambda equal to h upon p was obtained for photon. De Broglie said that if it is true for photon, it must be true for all the type of particles like electrons, neutrons and any other particle. So, this is a simple explanation given in support of De Broglie relation lambda equal to h upon p. Another justification for de Broglie hypothesis came from Bohr's postulates. According to Bohr's postulate, in an atom, electrons revolve around the nucleus in stationary orbits and in stationary orbits, the angular momentum of the electron is quantized. Angular momentum is integral multiple of h bar. Angular momentum is also given by mvr where m is mass of the electron, v is velocity with which the electron is moving and r is radius of stationary orbit. H bar is h upon 2 pi. So mvr is equal to nh upon 2 pi. This is what Bohr's postulate says. Now, if you consider de Broglie hypothesis which says there is a wave associated with the electron and wavelength of that wave is given by lambda equal to h upon p or lambda equal to h upon mv. Then, only those orbits will be selected by the electrons for which 2 pi r is equal to integral multiple of lambda. So, here you can see 2 pi r is not equal to integral multiple of lambda. So, this orbit will not be selected by the electron. But for this orbit, we can see 2 pi r, the circumference of this circular orbit is equal to integral multiple of lambda. In this case, it is 1, 2, 3, 4, 5, 5 lambda. So, this orbit can be selected by the electrons. So, if you accept the de Broglie relation then only those orbits will be selected for which the circumference of the circular orbit 2 pi r is integral multiple of lambda. That means 2 pi r is equal to n into h upon mv because according to de Broglie relation lambda is equal to h upon mv. Just take this MV on the left side and 2 pi on the right side. We can write it as MVR equal to NH upon 2 pi or MVR is equal to NH bar. This mvr equal to nh bar is nothing but Bohr's postulate. Angular momentum is quantized. So if we accept de Broglie relation lambda is equal to h upon mv and if we accept the fact that there is a wave associated with electron, then it arrives at the Bohr's postulate. mvr, the angular momentum is equal to nh bar. It means if we are accepting Bohr's postulate then there is no harm in accepting lambda is equal to h upon mv relation also. So, if Bohr's postulate is considered then de Broglie hypothesis can also be considered. So, this was a justification given to de Broglie hypothesis using Bohr's postulate. We can write de Broglie wavelength in terms of kinetic energy. De Broglie wavelength is given by lambda equal to h upon p that is equal to h upon mv where m is mass of the electron and v is velocity with which the electron is moving, h is Planck's constant. The kinetic energy of the particle is given by e is equal to 1 by 2 mv square which we can write as 1 by 2 m m square v square. just multiply and divide by m here. m square v square is nothing but p square. So, we can write from this p square is equal to 2me, 2m taken on the other side or p momentum of the particle to be equal to square root of 2me. So, we can just replace this p in the Broglie relation by square root of 2me and we can get the relation for lambda in terms of energy as lambda equal to h upon square root of 2me. If we are accelerating the electron by using potential difference of V volts, then this kinetic energy of the electron will be e into V where e is charge on the electron and therefore, lambda can be written as. h upon square root of 2 MeV. So, for a charged particle which is accelerated by using the potential difference of V volts, the wavelength of the wave associated with such charged particle will be equal to h upon square root of 2 MeV. And if E is the thermal energy, it can be written as kT where k is Boltzmann constant and T is temperature in kelvins. So, the wavelength of the wave associated will become h upon square root of 2 mkT. So, we can use these relations while solving the numerical problems. De Broglie proposed that as wave particle duality exist for photons that is light, the wave particle duality must exist for each and every type of particles. So, he said that There is a wave associated with every moving particle and wavelength of that wave is given by lambda equal to h upon p where p is momentum, h is the Planck's constant. We can divide this de Broglie hypothesis statement into two parts. First part says there is a wave associated with every moving particle and the second part says wavelength of that wave is given by lambda equal to h upon p. So, lambda equal to h upon p is called as de Broglie relation. The first part of the statement was justified by using Bohr's postulate. So, using Bohr's postulate, the existence of wave associated with the electron revolving in circular orbits inside the atom was justified. And the second part of the statement which says lambda equal to h upon p were justified by using Planck's relation E is equal to h nu and Einstein's relation E is equal to mc square. But both these justifications cannot be accepted as firm evidences because Bohr's postulate is a postulate, it is an assumption and lambda equal to h upon p de Broglie relation was obtained only in case of case of photon using E is equal to h nu and E is equal to mc square. So, some firm experimental evidence was needed so that de Broglie hypothesis can be accepted. The experimental evidence should show the existence of wave associated with some particle other than photon and the The wave associated with that particle should have the wavelength lambda equal to h upon p. So, number of attempts were made to find the experimental evidence which can show the wave associated with the particle and to calculate the wavelength of the wave associated with the particle. particle. Davison and Germer experiment was one of the attempts. So, let us discuss this Davison and Germer experiments and how the result in Davison-Germer experiment confirmed the existence of wave associated with the electron and how the results in this experiment confirmed the correctness of de Broglie relation lambda equal to h upon p. So here you can see the experimental setup of Davison and Germer experiment. In this experimental setup, you can see an electron gun. It consists of a tungsten filament. The electrons are emitted by the thermionic emission. When we pass current through the filament, it is heated and due to thermionic emission, the electrons are emitted. One anode is placed over here. This anode is kept at positive potential. So, this anode accelerates the beam of electrons in the downward direction. This anode is provided with one aperture in it so that a fine beam of electrons can be obtained. Now, the idea of the experiment is to get diffraction pattern by using the beam of electrons because if there is a wave associated with electron, that wave must produce the diffraction pattern and to produce the diffraction pattern. What we need is a diffraction grating. So here a target in the form of a nickel crystal is used as a diffraction grating. So here we are using the grating as a reflection grating. So nickel crystal is used as a diffraction grating. The electrons are made to fall on this nickel crystal. The electron beams will get reflected from different atomic planes inside the nickel crystal. Then one collector or detector was used to collect these electrons which are reflected from the nickel crystal. This collector will collect the electrons and these electrons are grounded through the galvanometer. So, this galvanometer will show us how many electrons are being collected. The current in the galvanometer will be proportional to the number of electrons that are collected by this collector. Now, there is a provision to move this collector and collect the number of electrons at different angles. So, this collector can be kept here and the number of electrons in this direction can be noted. It can be kept in this direction and the number of electrons in this particular direction. can be noted. For each position of the collector, the current in the galvanometer is noted. This current noted in the galvanometer is a measure of intensity of the diffracted beam of electrons. So, what is done is simply an electron beam is made incident on a nickel crystal which is used as a diffraction grating over here and the intensity of reflected beam is noted at various angles with the help of this collector and galvanometer. This procedure was repeated for different values of voltages maintained between the anode and the cathode. And a graph, polar graph was plotted between the current that is intensity of the electron beam and the angle between the incident and diffracted beam, this angle. This graph showed two peaks separated by a minimum. Two peaks separated by a minimum confirm the existence of wave associated with the electron because two peaks separated by minimum. means diffraction has taken place. Without diffraction, two peaks would not have been obtained. So, the existence of two peaks separated by a minimum confirm that diffraction has occurred and diffraction cannot be explained by using particle theory. So, if diffraction has occurred that means there must be a wave associated with the electrons. If wave is not associated with the electrons, then only one peak would have been obtained. But the existence of second peak indicates the diffraction has taken place and the diffraction has taken place means there is a wave associated with it. with the electrons because without waves diffraction cannot occur. So, this confirms the first part of the de Broglie hypothesis. With the electron, there is a wave associated. So, we have shown that for a particle other than photon also there is a wave associated. The experimental results show that for a voltage of around 54 volts, so when V is equal to 54 volts, the electrons scatter more prominently at an angle of 50 degrees with the direction of incident beam. So, we can say the first order maximum is obtained at an angle of 50 degrees. So, these values of V and phi will be using for calculating the value of wavelength. So, Davison-Germer obtained diffraction pattern with the help of beam of electrons. So, this confirms the first part of de Broglie hypothesis that there is a wave associated with moving electrons. Now, second part which says wavelength of the wave is given by lambda equal to h upon p. Now, to verify the correctness of this de Broglie relation, we can calculate this wavelength with electrons using this relation first and then we will calculate this wavelength by using some other known result in physics. If the two wavelengths match, then we can say the de Broglie relation is correct. So, let us first calculate this wavelength of the waves associated with electrons by using de Broglie relation. So, lambda is equal to h upon p, p is momentum, h is Planck's constant. So, p is a square root of 2me. So, e energy of the electrons here is e into v, v is the voltage, 54 volts we will use here because for 54 volts, the prominent maximum is obtained. so lambda is equal to h upon p that is equal to h upon square root of 2 m e v h is plank's constant which is 6.626 into 10 raise to minus 34 joule second m is mass of the electrons 9.1 into 10 raise to minus 31 kg E is charge on the electron 1.6 into 10 raise to minus 19 coulomb. So by using all these values the wavelength lambda is calculated and it was found to be 1.67 into 10 raise to minus 10 meters or 1.67. angstroms. So, using De Broglie relation, the wavelength is calculated, wavelength of the wave associated with electrons is calculated as 1.67 angstroms in this Davy-Zinn-Germer experiment. Now, let us calculate the wavelength of the wave associated with these electrons by using some other relation. Bragg's diffraction law. Bragg's diffraction law is already established law in physics. So if the wavelength calculated by using this Bragg's law of diffraction matches with de Broglie wavelength which is calculated as 1.67 angstroms, then we can say the de Broglie relation is correct. So let us calculate the wavelength by using Bragg's law of diffraction. So Bragg's law of diffraction is given as 2 d sin theta. equal to n lambda where d is the width of the slit. In this case, it will be the separation between the two atomic planes, gap between the two atomic planes. Theta is the glancing angle. So, theta is the glancing angle means it is the angle between the incident beam and the atomic plane. So, this is the angle theta. Now, what will be the value of this theta? This phi we have seen, the angle between the incident ray and the diffracted beam is 50 degrees. So, phi by 2 will be 25 degrees and therefore this theta, the glancing angle will be 90 minus 25 that is 65 degrees. So, we will substitute theta to be equal to 65 degrees. The inter planar spacing D for nickel crystal is 9.086 into 10 to the power minus 11 meters. So, D is used as 9.086 into 10 to the power minus 11 meters and The maximum is obtained for the first order. So, this n order of the maximum is taken as 1. Using these values, the wavelength lambda is calculated and it is found that the wavelength lambda is 1 by 2. 1.65 into 10 raise to minus 10 meters that is 1.65 Angstroms. So, just recollect the value calculated using de Broglie relation, it was 1.67 Angstroms. So, the two values calculated using de Broglie relation and Bragg's law of diffraction match. to a close proximity and therefore what we can say is if Bragg's law of diffraction is correct, then de Broglie relation must also be correct because it is also giving us almost same value as given by Bragg's law of diffraction. So, this is how Davy's and Germer experiment confirmed the second part of the de Broglie hypothesis also. So, in the first part we proved that there is a wave associated with electrons by obtaining the diffraction pattern and in the second part we proved the correctness of de Broglie relation by matching the wavelength obtained using de Broglie relation with the wavelength obtained by some other law which is Bragg's law of diffraction in physics. So, as the two values of wavelength match. If Bragg's law is correct then de Broglie relation is also correct and proved. So this is how Davison-Germer proved de Broglie hypothesis and de Broglie hypothesis was experimentally verified. But this de Broglie hypothesis in Davison-Germer experiment is experimentally verified only for the electrons which are very small particles which move with very high speed. Now, because in case of electron, the wavelength calculated is around 1.65 angstroms and the diffraction grating in which width of the slit that is this inter planar spacing of nickel crystal is of the same order. around 1 angstroms. Therefore, it was possible to perform this experiment and get the diffraction pattern because to perform diffraction experiment, the wavelength of the wave used in the experiment should be comparable to the width of the wave. the slit in the diffraction grating. Now if the wavelength is very very small for macroscopic bodies, if you calculate this wavelength, it will be very very small and the diffraction grating in which width of the slit is so small may not be available and therefore the existence of wave associated with macroscopic bodies, bigger bodies may not be possible. So, the experimental verification of the wave nature of macroscopic bodies may not be possible because diffraction grating in which width of the slit comparable to wavelength of the wave associated with such particles may not be available or is not available. So now it is It is confirmed that waves are associated with all the moving particles. So, such waves associated with moving particles are called as matter waves. Wavelength of the matter wave is given by lambda is equal to h upon p that is equal to h upon mv. This is called as de Broglie relation. These matter waves are not electromagnetic waves and can be associated with any particle whether the particle is charged or uncharged. The matter waves can propagate in vacuum and hence they are not mechanical waves. If you calculate the phase velocity of the matter wave, it will be c square upon v that is greater than c. So, we will talk on this phase velocity in our next two sessions. So, for the moment just keep in mind the phase velocity can be greater than c. So, these are the properties of matter waves. So, we are at the end of session 1 on quantum physics. In this session, we discussed de Broglie hypothesis. We stated the de Broglie hypothesis. We have seen how this de Broglie hypothesis was justified by using Bohr's postulates. Then, we have seen how Planck's relation and Einstein relation was used to justify de Broglie relation lambda is equal to h upon p and at the end, we have seen how Davyzin and German experimentally verified the correctness of de Broglie hypothesis. In the immediately next session, we will be solving some numerical problems based on de Broglie hypothesis and in the coming sessions, we will be discussing Heisenberg uncertainty principle and Schrodinger equations. So, I hope you must have liked this video. If you liked this video, then you can subscribe to my channel so that you can get timely intimation of new videos that will be uploaded to my channel. So, thank you. Thank you once again for watching.

Hello, welcome once again to my engineering physics channel. This is Sanju Badhe from K.J. Somaya Institute of Engineering and Information Technology.

From this session, we are starting with discussion on quantum physics. In the first session, we will be getting introduced to De Broglie hypothesis and in further sessions, we will be studying Heisenberg uncertainty principle and Schrodinger's equations. This is the flow of discussion that we will be following for the discussion.

First, we will get introduced to de Broglie hypothesis of matter waves. We will discuss properties of the matter waves. Then we will get introduced to the concept of wave packet Heisenberg and uncertainty principle, one important application of Heisenberg uncertainty principle which explains why electron cannot exist in the nucleus. Then we will derive Schrodinger's time dependent equation and Schrodinger's time independent equation and we will consider a case of a particle trapped in one dimensional infinite potential well. And at the end, we will be discussing application of quantum physics, one important application quantum coding.

The concept of quantum physics was first used by Max Planck to explain the results obtained in black body radiation experiment. A black body is designed to absorb 99.9 percent of the radiation incident on it. Now, how this black body can be obtained?

This was done with a hollow sphere with a narrow hole. Inner surface of this hollow sphere is coated with carbon black. When the radiation enters through a narrow hole, it hits the wall of the sphere and gets almost fully absorbed. Remaining radiation is reflected back and collide again with the wall.

After multiple reflections, almost all the radiation is absorbed by this black body. Due to the absorption of radiation, temperature of the body increases and then it starts emitting the radiation from the narrow hole. This radiation contains all the wavelengths. This is called as the black body radiation. When intensity versus wavelength curve is plotted, it is found that radiation for higher wavelength is very low and for shorter wavelength the intensity is more.

For a particular wavelength called lambda max at a given temperature, the intensity is maximum. The wavelength for which maximum energy is radiated is called as the peak wavelength. As you can see in the graph, all the wavelengths are emitted but there is maximum intensity only for the peak wavelength.

The black body will appear of the color corresponding to that peak wavelength. If you increase the temperature, what happens to this peak wavelength? This peak wavelength is shifting to the left. So, for temperature equal to 3000 Kelvin, peak wavelength is somewhere around here.

For 4000 Kelvin, the peak wavelength shifts to the left. left side. For 5000 Kelvin, again the peak wavelength shifts to the left side.

For 6000, it further shifts to the left side. So, if we increase the temperature of the black body, the lambda max, the peak wavelength shift towards the left side and the color of the black body will change accordingly. This shift in the peak wavelength was explained by Vance displacement law which states that the peak wavelength is inversely proportional to the temperature.

So as the temperature increases, the peak wavelength will shift towards the left side. As the temperature increases, the peak wavelength will reduce. So, this Vance displacement law could explain the shift in the peak wavelength as the temperature increases but this law could not explain the part of the curve towards the longer wavelength.

So, for higher wavelengths, this law could not explain the behavior of the curve. Also, if the temperature is very close to 0, then the peak wavelength will become infinite. But when the temperature is lowered, the radiation curve changes the behavior which also could not be explained by this Vann's law. Rayleigh and Zinn tried to explain the blackbody radiation curve using their Rayleigh-Zinn's radiation law which is based on classical statistics. According to this law, the energy density or the intensity is directly proportional to square of the frequency of radiation.

That means it is inversely proportional to square of the wavelength. This Rayleigh-Zinn's radiation law could explain the behavior of this radiation curve. radiation curve on the higher wavelength side.

So, for higher wavelengths it is applicable, but when it comes to the lower wavelength, this law could not explain the behavior of the radiation curve. So, Vann's law could explain the part of the curve on the lower wavelength side and Rayleigh-Jeans radiation law could explain the part of this radiation curve, black body radiation curve on the higher wavelength side. But the complete curve could not be explained by any of these two laws.

So, we have seen Vance displacement law could explain only the shorter wavelength side of this blackbody radiation curve and Rayleigh-Jeans law could explain only the longer wavelength side of this blackbody radiation curve. The entire curve could not be explained. explained by any of these two laws.

To explain the black body radiation curve, Max Planck came forward and he suggested that radiated energy must be depending on frequency of radiation and he represented this energy radiated E to be equal to h nu where h is called as Planck's constant. Max Planck calculated the average energy of radiation using Maxwell Boltzmann distribution law and this energy was calculated as E average equal to h nu upon E raise to h nu by kT minus 1. Max Planck simply replaced the average energy kT in Rayleigh-Jeans formula by this new value calculated by him and the Rayleigh-Jeans formula now becomes rho V, the energy density density is equal to 8 pi nu square upon c cube as it is you can see into the average value calculated by max Planck, average energy value calculated by max Planck h nu upon e raise to h nu by kT minus 1. So, this will become rho v equal to 8 pi h nu cube upon c cube into bracket 1 upon e raise to h nu by kT minus 1. This is called as Planck's radiation law and this Planck's radiation law now gives correct explanation for the blackbody radiation curve. So, while explaining the black body radiation curve, Planck has introduced this revolutionary idea which says energy of the radiation depends on the frequency and is given by E is equal to h nu.

Einstein used this idea to explain his photoelectric effect. This E is equal to h nu is called as quantum of light energy. Max Planck was awarded Nobel Prize in in 1918 for this discovery. Einstein used Planck's idea which says energy radiated must be depending on the frequency of radiation and E is equal to h nu. Einstein confirmed that light consists of discrete unit of energy which are known as photons and each photon has energy E is equal to h nu where nu is frequency of the radiation.

This photoelectric effect confirmed the dual nature of light. So, light behaves as particle as well as wave. As far as macroscopic particles which are bigger in size, moving with comparatively lower speeds are considered, the motion of such particles can be explained by classical theory of mechanics. But when it comes to microscopic particles like electrons, protons, etc which move with very high speed, this classical theory of mechanics fails to explain the motion of such particle. So, it was needed to develop a new theory which could explain the motion of such microscopic particles.

So, quantum mechanics was developed from quantum theory to explain the properties associated with such particles. De Broglie wave particle duality hypothesis, Heisenberg uncertainty principle and Schrodinger equation provide base on which the quantum mechanics is built. So, we will be discussing this De Broglie hypothesis, Heisenberg uncertainty principle and Schrodinger's equation one by one in the coming sessions. So, we have seen photoelectric effect introduced firm evidence of a particle nature of light. Data There are some phenomena like interference, diffraction, polarization which could be explained by considering only wave nature of light.

But photoelectric effect, to explain the photoelectric effect, particle nature has to be considered. It shows the light exhibits a dual nature. Deeprogli just took the same idea. He said if dual nature exists for light, the dual nature must exist for all type of matter.

Because the nature is symmetric, if one phenomenon is applicable to certain type of matter, then it is called as dual nature. matter, the same phenomenon should be applicable to all type of matter. And he stated his hypothesis as there is a wave associated with every moving particle moving with velocity v and the wavelength this wave is given by lambda is equal to h upon p that is h upon mv where m is mass of the particle, v is velocity with which the particle is moving and h is Planck's constant. So according to de Broglie hypothesis, every moving particle has a wave associated with it.

This is the first part and wavelength of this wave is given by lambda is equal to h upon p, this is the second part of the hypothesis. Why it is called as hypothesis? Because the firm evidence was not available. It is just a proposed explanation made on the basis of some limited evidence as a starting point for further investigation.

De Broglie just said if wave particle duality exist for light, then wave particle duality must exist for all type of particles. So, there should be a wave associated with electron also. There should be a wave associated with any other particle which is moving with certain velocity v and wavelength of that wave should be lambda equal to h upon p.

So, this is what the Broglie hypothesis says. De Broglie relation can be supported by using Planck's theory and Einstein's theory. Energy of a photon according to Planck's theory is given by E is equal to h nu where h is Planck's constant and nu is frequency of radiation. And if we consider photon as a particle of mass m, then its energy as per Einstein's mass energy relation is given as E is equal to mc square. So, both h nu and mc square are energies of the photon and therefore we can equate them.

h nu equal to mc square, let us call this as equation number 3. The photon is travelling with velocity c in free space. So, velocity of light is c. So, photon will travel with velocity equal to c. So, its momentum p is given by m into c.

So, momentum of photon is equal to m into c. Let us call this as equation number 4. De Broglie supported his hypothesis by using Planck's relation E is equal to h nu and Einstein's mass energy relation E is equal to mc square. So h nu and mc square both are energies of the photon. So h nu can be equated with mc square. The momentum of the photon P is equal to mc mass into velocity.

So, we can divide equation number 3 by equation number 4. So, h nu by p is equal to mc square upon mc. So, 1c will get cancelled here, mm will get cancelled with each other. So, h nu upon p is equal to c or h upon p is equal to c by nu and c by nu is nothing but lambda, the wavelength.

So, lambda is equal to h upon p. So, de Broglie just said if E is equal to h nu that is Planck's relation is correct and E is equal to mc square that is Einstein's relation is correct then lambda must be equal to h upon p. So, this lambda equal to h upon p was obtained for photon.

De Broglie said that if it is true for photon, it must be true for all the type of particles like electrons, neutrons and any other particle. So, this is a simple explanation given in support of De Broglie relation lambda equal to h upon p. Another justification for de Broglie hypothesis came from Bohr's postulates.

According to Bohr's postulate, in an atom, electrons revolve around the nucleus in stationary orbits and in stationary orbits, the angular momentum of the electron is quantized. Angular momentum is integral multiple of h bar. Angular momentum is also given by mvr where m is mass of the electron, v is velocity with which the electron is moving and r is radius of stationary orbit. H bar is h upon 2 pi.

So mvr is equal to nh upon 2 pi. This is what Bohr's postulate says. Now, if you consider de Broglie hypothesis which says there is a wave associated with the electron and wavelength of that wave is given by lambda equal to h upon p or lambda equal to h upon mv.

Then, only those orbits will be selected by the electrons for which 2 pi r is equal to integral multiple of lambda. So, here you can see 2 pi r is not equal to integral multiple of lambda. So, this orbit will not be selected by the electron.

But for this orbit, we can see 2 pi r, the circumference of this circular orbit is equal to integral multiple of lambda. In this case, it is 1, 2, 3, 4, 5, 5 lambda. So, this orbit can be selected by the electrons.

So, if you accept the de Broglie relation then only those orbits will be selected for which the circumference of the circular orbit 2 pi r is integral multiple of lambda. That means 2 pi r is equal to n into h upon mv because according to de Broglie relation lambda is equal to h upon mv. Just take this MV on the left side and 2 pi on the right side.

We can write it as MVR equal to NH upon 2 pi or MVR is equal to NH bar. This mvr equal to nh bar is nothing but Bohr's postulate. Angular momentum is quantized.

So if we accept de Broglie relation lambda is equal to h upon mv and if we accept the fact that there is a wave associated with electron, then it arrives at the Bohr's postulate. mvr, the angular momentum is equal to nh bar. It means if we are accepting Bohr's postulate then there is no harm in accepting lambda is equal to h upon mv relation also.

So, if Bohr's postulate is considered then de Broglie hypothesis can also be considered. So, this was a justification given to de Broglie hypothesis using Bohr's postulate. We can write de Broglie wavelength in terms of kinetic energy.

De Broglie wavelength is given by lambda equal to h upon p that is equal to h upon mv where m is mass of the electron and v is velocity with which the electron is moving, h is Planck's constant. The kinetic energy of the particle is given by e is equal to 1 by 2 mv square which we can write as 1 by 2 m m square v square. just multiply and divide by m here.

m square v square is nothing but p square. So, we can write from this p square is equal to 2me, 2m taken on the other side or p momentum of the particle to be equal to square root of 2me. So, we can just replace this p in the Broglie relation by square root of 2me and we can get the relation for lambda in terms of energy as lambda equal to h upon square root of 2me. If we are accelerating the electron by using potential difference of V volts, then this kinetic energy of the electron will be e into V where e is charge on the electron and therefore, lambda can be written as.

h upon square root of 2 MeV. So, for a charged particle which is accelerated by using the potential difference of V volts, the wavelength of the wave associated with such charged particle will be equal to h upon square root of 2 MeV. And if E is the thermal energy, it can be written as kT where k is Boltzmann constant and T is temperature in kelvins.

So, the wavelength of the wave associated will become h upon square root of 2 mkT. So, we can use these relations while solving the numerical problems. De Broglie proposed that as wave particle duality exist for photons that is light, the wave particle duality must exist for each and every type of particles. So, he said that There is a wave associated with every moving particle and wavelength of that wave is given by lambda equal to h upon p where p is momentum, h is the Planck's constant.

We can divide this de Broglie hypothesis statement into two parts. First part says there is a wave associated with every moving particle and the second part says wavelength of that wave is given by lambda equal to h upon p. So, lambda equal to h upon p is called as de Broglie relation. The first part of the statement was justified by using Bohr's postulate.

So, using Bohr's postulate, the existence of wave associated with the electron revolving in circular orbits inside the atom was justified. And the second part of the statement which says lambda equal to h upon p were justified by using Planck's relation E is equal to h nu and Einstein's relation E is equal to mc square. But both these justifications cannot be accepted as firm evidences because Bohr's postulate is a postulate, it is an assumption and lambda equal to h upon p de Broglie relation was obtained only in case of case of photon using E is equal to h nu and E is equal to mc square. So, some firm experimental evidence was needed so that de Broglie hypothesis can be accepted.

The experimental evidence should show the existence of wave associated with some particle other than photon and the The wave associated with that particle should have the wavelength lambda equal to h upon p. So, number of attempts were made to find the experimental evidence which can show the wave associated with the particle and to calculate the wavelength of the wave associated with the particle. particle. Davison and Germer experiment was one of the attempts. So, let us discuss this Davison and Germer experiments and how the result in Davison-Germer experiment confirmed the existence of wave associated with the electron and how the results in this experiment confirmed the correctness of de Broglie relation lambda equal to h upon p.

So here you can see the experimental setup of Davison and Germer experiment. In this experimental setup, you can see an electron gun. It consists of a tungsten filament. The electrons are emitted by the thermionic emission. When we pass current through the filament, it is heated and due to thermionic emission, the electrons are emitted.

One anode is placed over here. This anode is kept at positive potential. So, this anode accelerates the beam of electrons in the downward direction. This anode is provided with one aperture in it so that a fine beam of electrons can be obtained. Now, the idea of the experiment is to get diffraction pattern by using the beam of electrons because if there is a wave associated with electron, that wave must produce the diffraction pattern and to produce the diffraction pattern.

What we need is a diffraction grating. So here a target in the form of a nickel crystal is used as a diffraction grating. So here we are using the grating as a reflection grating. So nickel crystal is used as a diffraction grating.

The electrons are made to fall on this nickel crystal. The electron beams will get reflected from different atomic planes inside the nickel crystal. Then one collector or detector was used to collect these electrons which are reflected from the nickel crystal. This collector will collect the electrons and these electrons are grounded through the galvanometer. So, this galvanometer will show us how many electrons are being collected.

The current in the galvanometer will be proportional to the number of electrons that are collected by this collector. Now, there is a provision to move this collector and collect the number of electrons at different angles. So, this collector can be kept here and the number of electrons in this direction can be noted.

It can be kept in this direction and the number of electrons in this particular direction. can be noted. For each position of the collector, the current in the galvanometer is noted.

This current noted in the galvanometer is a measure of intensity of the diffracted beam of electrons. So, what is done is simply an electron beam is made incident on a nickel crystal which is used as a diffraction grating over here and the intensity of reflected beam is noted at various angles with the help of this collector and galvanometer. This procedure was repeated for different values of voltages maintained between the anode and the cathode.

And a graph, polar graph was plotted between the current that is intensity of the electron beam and the angle between the incident and diffracted beam, this angle. This graph showed two peaks separated by a minimum. Two peaks separated by a minimum confirm the existence of wave associated with the electron because two peaks separated by minimum. means diffraction has taken place.

Without diffraction, two peaks would not have been obtained. So, the existence of two peaks separated by a minimum confirm that diffraction has occurred and diffraction cannot be explained by using particle theory. So, if diffraction has occurred that means there must be a wave associated with the electrons. If wave is not associated with the electrons, then only one peak would have been obtained. But the existence of second peak indicates the diffraction has taken place and the diffraction has taken place means there is a wave associated with it.

with the electrons because without waves diffraction cannot occur. So, this confirms the first part of the de Broglie hypothesis. With the electron, there is a wave associated.

So, we have shown that for a particle other than photon also there is a wave associated. The experimental results show that for a voltage of around 54 volts, so when V is equal to 54 volts, the electrons scatter more prominently at an angle of 50 degrees with the direction of incident beam. So, we can say the first order maximum is obtained at an angle of 50 degrees.

So, these values of V and phi will be using for calculating the value of wavelength. So, Davison-Germer obtained diffraction pattern with the help of beam of electrons. So, this confirms the first part of de Broglie hypothesis that there is a wave associated with moving electrons.

Now, second part which says wavelength of the wave is given by lambda equal to h upon p. Now, to verify the correctness of this de Broglie relation, we can calculate this wavelength with electrons using this relation first and then we will calculate this wavelength by using some other known result in physics. If the two wavelengths match, then we can say the de Broglie relation is correct. So, let us first calculate this wavelength of the waves associated with electrons by using de Broglie relation. So, lambda is equal to h upon p, p is momentum, h is Planck's constant.

So, p is a square root of 2me. So, e energy of the electrons here is e into v, v is the voltage, 54 volts we will use here because for 54 volts, the prominent maximum is obtained. so lambda is equal to h upon p that is equal to h upon square root of 2 m e v h is plank's constant which is 6.626 into 10 raise to minus 34 joule second m is mass of the electrons 9.1 into 10 raise to minus 31 kg E is charge on the electron 1.6 into 10 raise to minus 19 coulomb.

So by using all these values the wavelength lambda is calculated and it was found to be 1.67 into 10 raise to minus 10 meters or 1.67. angstroms. So, using De Broglie relation, the wavelength is calculated, wavelength of the wave associated with electrons is calculated as 1.67 angstroms in this Davy-Zinn-Germer experiment.

Now, let us calculate the wavelength of the wave associated with these electrons by using some other relation. Bragg's diffraction law. Bragg's diffraction law is already established law in physics.

So if the wavelength calculated by using this Bragg's law of diffraction matches with de Broglie wavelength which is calculated as 1.67 angstroms, then we can say the de Broglie relation is correct. So let us calculate the wavelength by using Bragg's law of diffraction. So Bragg's law of diffraction is given as 2 d sin theta. equal to n lambda where d is the width of the slit. In this case, it will be the separation between the two atomic planes, gap between the two atomic planes.

Theta is the glancing angle. So, theta is the glancing angle means it is the angle between the incident beam and the atomic plane. So, this is the angle theta.

Now, what will be the value of this theta? This phi we have seen, the angle between the incident ray and the diffracted beam is 50 degrees. So, phi by 2 will be 25 degrees and therefore this theta, the glancing angle will be 90 minus 25 that is 65 degrees. So, we will substitute theta to be equal to 65 degrees.

The inter planar spacing D for nickel crystal is 9.086 into 10 to the power minus 11 meters. So, D is used as 9.086 into 10 to the power minus 11 meters and The maximum is obtained for the first order. So, this n order of the maximum is taken as 1. Using these values, the wavelength lambda is calculated and it is found that the wavelength lambda is 1 by 2. 1.65 into 10 raise to minus 10 meters that is 1.65 Angstroms.

So, just recollect the value calculated using de Broglie relation, it was 1.67 Angstroms. So, the two values calculated using de Broglie relation and Bragg's law of diffraction match. to a close proximity and therefore what we can say is if Bragg's law of diffraction is correct, then de Broglie relation must also be correct because it is also giving us almost same value as given by Bragg's law of diffraction.

So, this is how Davy's and Germer experiment confirmed the second part of the de Broglie hypothesis also. So, in the first part we proved that there is a wave associated with electrons by obtaining the diffraction pattern and in the second part we proved the correctness of de Broglie relation by matching the wavelength obtained using de Broglie relation with the wavelength obtained by some other law which is Bragg's law of diffraction in physics. So, as the two values of wavelength match. If Bragg's law is correct then de Broglie relation is also correct and proved. So this is how Davison-Germer proved de Broglie hypothesis and de Broglie hypothesis was experimentally verified.

But this de Broglie hypothesis in Davison-Germer experiment is experimentally verified only for the electrons which are very small particles which move with very high speed. Now, because in case of electron, the wavelength calculated is around 1.65 angstroms and the diffraction grating in which width of the slit that is this inter planar spacing of nickel crystal is of the same order. around 1 angstroms.

Therefore, it was possible to perform this experiment and get the diffraction pattern because to perform diffraction experiment, the wavelength of the wave used in the experiment should be comparable to the width of the wave. the slit in the diffraction grating. Now if the wavelength is very very small for macroscopic bodies, if you calculate this wavelength, it will be very very small and the diffraction grating in which width of the slit is so small may not be available and therefore the existence of wave associated with macroscopic bodies, bigger bodies may not be possible.

So, the experimental verification of the wave nature of macroscopic bodies may not be possible because diffraction grating in which width of the slit comparable to wavelength of the wave associated with such particles may not be available or is not available. So now it is It is confirmed that waves are associated with all the moving particles. So, such waves associated with moving particles are called as matter waves.

Wavelength of the matter wave is given by lambda is equal to h upon p that is equal to h upon mv. This is called as de Broglie relation. These matter waves are not electromagnetic waves and can be associated with any particle whether the particle is charged or uncharged.

The matter waves can propagate in vacuum and hence they are not mechanical waves. If you calculate the phase velocity of the matter wave, it will be c square upon v that is greater than c. So, we will talk on this phase velocity in our next two sessions.

So, for the moment just keep in mind the phase velocity can be greater than c. So, these are the properties of matter waves. So, we are at the end of session 1 on quantum physics. In this session, we discussed de Broglie hypothesis. We stated the de Broglie hypothesis.

We have seen how this de Broglie hypothesis was justified by using Bohr's postulates. Then, we have seen how Planck's relation and Einstein relation was used to justify de Broglie relation lambda is equal to h upon p and at the end, we have seen how Davyzin and German experimentally verified the correctness of de Broglie hypothesis. In the immediately next session, we will be solving some numerical problems based on de Broglie hypothesis and in the coming sessions, we will be discussing Heisenberg uncertainty principle and Schrodinger equations.

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Transcript for:Overview of Quantum Physics Concepts

Transcript for:
Overview of Quantum Physics Concepts