beginner stuff

Question 1

equation of a plane wave

Answer 1

Φ(x,t) = A * e^i(kx - ωt)

k: wave number
A: normalisation constant
ω: angular freq

Question 2

relate E and ω

Answer 2

ω = 2π*f = 2π E/h = E/ℏ

Question 3

ω and k relation

Answer 3

ωℏ = ℏ²k²/2m

Question 4

construct the wave equation from the ω and k relation

Answer 4

if Φ(x,t) = A * e^i(kx - ωt)

∂²Φ/∂x² = -k²Φ and ∂Φ/∂t = -iωΦ

from the ω and k relation:
ℏ²k²/2m * Φ = ωℏ * Φ

hence:
ℏ²/2m * ∂²Φ/∂x² = iℏ * ∂Φ/∂t

aka TDSE

Question 5

what does it mean for a wave function to be normalised?

Answer 5

∫|Ψ|² dx = 1

over all space

Question 6

properties of the wavefunction

Answer 6

1. Normalised
2. continuous
3. smooth (continuous derivative)
4. fragile (changes when measured)

2+3 are only for a none infinite potential well

Question 7

derive the TISE using variable separation

Answer 7

assume Ψ(x, t) = ψ(x)T(t)

sub into ℏ²/2m * ∂²Ψ/∂x² = iℏ * ∂Ψ/∂t
gives T * ℏ²/2m * ∂²ψ/∂x² = ψ * iℏ * ∂T/∂t
divide by Ψ:
1/ψ * ℏ²/2m * ∂²ψ/∂x² = 1/T * iℏ * ∂T/∂t

since a function only of x equals a function only of t they must equal a constant. it has units of energy so its E

ℏ²/2m * ∂²ψ/∂x² = Eψ

Question 8

derive the general solution to the TDSE

Answer 8

when deriving the TISE you get
ℏ²/2m * ∂²ψ/∂x² = Eψ
but also:
iℏ * ∂T/∂t = E * T(t)

∂T/∂t = -iE/ℏ * T(t)
T = A * e^-iEt/ℏ

hence the general solution to Ψ is:
Ψ(x, t) = Σᵢ Aᵢψᵢe^-iEt/ℏ