Quantum

Question 1

Schrodinger equation

Answer 1

iħ∂Ψ/∂t = -(ħ²/2m)∂²Ψ/∂x² + VΨ

• V -> potential energy function*
• m -> mass of particle*

Question 2

probability and wave function

Answer 2

∫|Ψ(x,t)|²dx = probability of finding particle at area under curve

Question 3

uncertainty principle

Answer 3

σ_xσ_p ≥ ħ/2

• σ_x -> uncertainty in position*
• σ_p -> uncertainty in momentum*

Question 4

requirements for orthognal wave function

Answer 4

∫Ψ_m(x)*Ψ_n(x)dx = 0

if m != n

Question 5

requirements for normalized wave function

Answer 5

∫Ψ_m(x)*Ψ_n(x)dx = δ_mn

δ_mn -> Kronickers delta -> 0 if m != n, 1 if m = n

Question 6

solution for quantum square well

Answer 6

Ψ_n(x) = √(2/a)sin(nπx/a)

• n = 1,2,3 …*
• a -> length of square well*
• with increasing n, a node is added to wave function*

Question 7

steps to find A (normalize wave function)

Answer 7

∫|Ψ(x, 0)|²dx = 1

limits are limits of well

Question 8

energies for infinite square well

Answer 8

E_n = n²π²ħ²/2ma²

• a -> length of square well*
• n -> 1,2,3 …*

Question 9

energy levels for quantum harmonic oscillator

Answer 9

En = (n + 1/2)ħω

• ω -> √(k/m) *
• k -> bound force constant*
• m -> reduced mass m₁m₂/(m₁ + m₂)*

Question 10

commutor opperations

Answer 10

[X,Y] = Z
[XY,Z] = X[Y,Z] + [X,Z]Y