2 Schrödinger equation as description of QM systems

Question 1

What is the 1d Schrödinger equation?

Answer 1

iℏ ∂ψ(x,t)/∂t = -ℏ²/2M ∂²ψ(x,t)/∂x²+V(x,t)ψ(x,t)

Question 2

What is the momentum operator?

Answer 2

-iℏ ∂/∂x

Question 3

What is the energy operator?

Answer 3

iℏ ∂/∂t

Question 4

What are the terms in the Schr equation?

Answer 4

The kinetic and potential energy

Question 5

What is the probability of finding a quantum particle in an interval?

Answer 5

P(x,t)dx = |ψ(x,t)|² = ψ^*(x,t)ψ(x,t)
The Born postulate

Question 6

What is the normalisation of a wave function?

Answer 6

∫ψ^*ψdx = 1

Question 7

What is the solution for the TISE?

Answer 7

ψ(x,t) = T(t)Φ(x)

Question 8

What are the independent equations for the TISE?

Answer 8

E= iℏ 1/T(t) dT(t)/dt
E= -ℏ²/2M 1/Φ(x) d²Φ(x)/dx² +V(x)

Question 9

What is the wave function for the TISE?

Answer 9

ψ(x,t) = Φ(x)e^-iEt/ℏ

Question 10

What are the boundary conditions for the wave function?

Answer 10

1. The function has to be a continuous, single-valued function for all positions and times.
2. The integral of the squared modulus of the wave function over the whole space has to be finite. This ensures that the wave function can be normalised.
3. The first derivative of the wave function has to be continuous everywhere except where the potential has an infinite step.

Question 11

What are the purposes of the boundary conditions?

Answer 11

To have a defined probability
To normalise the wave function
To allow it to be a solution of the Schr equation

Question 12

What is the form of the oscillating solutions?

Answer 12

Φ(x) = Acos(kx) + Bsin(kx)

Question 13

What are the energy levels for wave functions?

Answer 13

E_n = ℏ²π²n²/8Ma²

Question 14

What is the solution on the left side of a potential barrier?

Answer 14

Φ₁ = Ae^ikx+Be^-ikx

Question 15

What is the solution on the right side of a potential barrier?

Answer 15

Φ₃ = Fe^ikx

Question 16

What is the solution on the middle side of a potential barrier?

Answer 16

Φ₂ = Ce^κx +De^-κx
Where κ = sqrt(2m(V₀-E)/ℏ²)

Question 17

What is the transmission probability for quantum tunnelling?

Answer 17

T = |F|²/|A|²

Question 18

What is the potential energy of a spring?

Answer 18

V(x) = kx²/2

Question 19

What is the TISE equation for the harmonic oscillator?

Answer 19

-ℏ²/2m ∂²Φ(x)/∂x² + mω²x²Φ(x)/2 = EΦ(x)
Where ω = sqrt(k/M)

Question 20

What are the energy levels for a harmonic oscillator?

Answer 20

E = (1/2 + n) ℏω