Lecture Notes on the Hydrogen Atom and Quantum Numbers
Bohr Model vs. Quantum Mechanics
Bohr Model:
Describes the hydrogen atom with one electron in a fixed orbit around the nucleus.
Treats the electron similarly to planets orbiting the sun.
Quantum Mechanics:
Does not define a precise location for the electron.
Introduces the concept of an orbital: a region of space where the electron is likely to be found, often visualized as a sphere around the nucleus.
Quantum mechanics is a more accurate representation of electron behavior than the Bohr model.
Four Quantum Numbers
Principal Quantum Number (n):
Symbol: n
Positive integer (n = 1, 2, 3, ...)
Indicates the main energy level or shell occupied by the electron.
Higher n values mean greater average distance from the nucleus and higher energy.
Example:
n = 1: electron closer to the nucleus.
n = 2: electron further away with higher energy.
Angular Momentum Quantum Number (l):
Symbol: l
Describes the shape of the orbital.
Values range from 0 to (n - 1).
Example:
n = 1: l = 0 (s orbital, spherical shape).
n = 2: l = 0 (s orbital) and l = 1 (p orbital, dumbbell shape).
Magnetic Quantum Number (m_l):
Symbol: m_l
Indicates the orientation of the orbital in space.
Values range from -l to +l.
Example for l = 0 (s orbital): only 1 orientation (m_l = 0).
Example for l = 1 (p orbital): three orientations (m_l = -1, 0, +1).
p orbitals are oriented along the x, y, and z axes (px, py, pz).
Spin Quantum Number (m_s):
Symbol: m_s
Represents the intrinsic spin of the electron (not literally spinning).
Two possible values:
+1/2 (spin up)
-1/2 (spin down)
Visualized as arrows pointing up or down.
Summary
The four quantum numbers (n, l, m_l, m_s) together describe the unique state of an electron in an atom, allowing for a deeper understanding of electron configuration and behavior in quantum mechanics.