New Quantum Mechanics
Operator
In QM an observable (like position, momentum, energy, etc.) is represented by an
෡)
operator.(in general 𝑨
Classical Phys: (just a
number)
𝑥 (the coordinate of the
particle)
QM (operator acting on wave
function)
𝑥ො = 𝑥 (multiplies the wave function
by 𝑥)
Momentum
𝑝 = 𝑚𝑣
Energy (Total)
𝑝2
𝐸=
+𝑈 𝑥
2𝑚
𝜕
𝑝Ƹ = −𝑖ℏ
𝜕𝑥
ℏ2 𝜕 2
෡=−
𝐻
+ 𝑈(𝑥)
2𝑚 𝜕𝑥 2
𝐿 = 𝑟Ԧ × 𝑝Ԧ
෠𝐿 = −𝑖ℏ 𝜕
Exact value
Only eigenvalue of the operator
Newton’s laws (𝐹 = 𝑚𝑎)
SE (𝑖ℏ
Observable
Position
Angular
momentum
Measurement
Outcome
Evolution
𝜕∅
𝜕
෡ 𝜓)
𝜓= 𝐻
𝜕𝑡
The Schrodinger Equation (Law of Psi)
 Newton’s laws are the laws of classical mechanics.
 Schrodinger equation is the basic law of quantum mechanics.
 If you know a particle’s wave function, you can predict the probability of detecting it in
some region of space.
 Classical systems are described in terms of forces.
 In contrast, a quantum system is described by a potential energy function 𝑼(𝒙) that
describes a particle’s interactions with its environment.
Time-dependent Schrodinger equation for one dimension (TDSE)
𝜕𝜓(𝑥, 𝑡)
ℏ2 𝜕 2 𝜓 𝑥, 𝑡
𝑖ℏ
= −
+ 𝑈(𝑥)𝜓(𝑥, 𝑡)
2
𝜕𝑡
2𝑚
𝜕𝑥
𝜕𝜓
෡𝜓
𝑖ℏ
= 𝐻
𝜕𝑡
෡ is called the Hamiltonian operator, operating on 𝜙 that represent the total energy of the
𝐻
particle. (Energy operator)
Probability Density(PD) or Probability Density Function(PDF)
∞
𝑝 𝑥, 𝑡 = න 𝜓(𝑥, 𝑡) 2 𝑑𝑥
−∞
Normalization:
∞
‫׬‬−∞ 𝜓(𝑥, 𝑡) 2 𝑑𝑥 = 1
Expectation Values
∞
∞
𝐴 = න 𝐴መ 𝜓 2 𝑑𝑥 = න 𝜓 ∗ 𝐴መ 𝜓 𝑑𝑥
−∞
−∞
Time-Independent Schrodinger’s Equation (TISE)
ℏ2 𝜕 2 𝜙
−
+ 𝑈 𝜙=𝐸 𝜙
2
2𝑚 𝜕𝑥
෡𝜙=𝐸 𝜙
𝐻
Energy Levels for a particle in a box (Infinite Potential Well )
𝑬𝒏 =
𝒏𝝅ℏ 𝟐
𝒏𝒉 𝟐
=
𝟐
𝟐𝒎𝑳
𝟖𝒎𝑳𝟐
𝑛 = 1,2,3, …
The boundary condition of the wave function,
𝜓 0 = 𝜓 𝐿 = 0
For the particle is free inside the box (free particle), 𝑼 𝒙 = 𝟎 (𝒊𝒏𝒔𝒊𝒅𝒆 𝒕𝒉𝒆 𝒃𝒐𝒙)
0
𝑈 𝑥 = ቊ
∞
0<𝑥<𝐿
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
The wave function corresponding to nth energy level
𝒏𝝅𝒙
𝝍𝒏 𝒙 = 𝑨 𝒔𝒊𝒏
𝑳
 TISE gives discrete energy levels / allowed energies and wave functions.