Quantum Theory

Question 1

Quantum Theory

What are the De Broglie relations and when do they apply (D)

Answer 1

E=hw (Energy, constant, angular frequency)
p=hk (Momentum vector, constant, wave vector)

They apply for a free particle (no forces / V=0)

Question 2

Quantum Theory

What is the Schrodinger equation? (D)

Answer 2

ih * ∂𝛹/∂t = - h^2 / 2m * ∇^2𝛹 + V𝛹

Question 3

Quantum Theory

What is the stationary state Schrodinger equation? (D)

Answer 3

• h^2 / 2m * ∇^2𝜓+V𝜓=E𝜓

Question 4

Quantum Theory

How do you arrive at the stationary state equation from the Schrodinger equation? (Q)

Answer 4

You look for separable solutions. Let 𝛹(x,t)=𝜓(x)T(t) and divide by it to get:

( ih dT/dt ) / T
= ( - h^2 / 2m * ∇^2𝜓 + V𝜓 ) / 𝜓
= constant
= E

Question 5

Quantum Theory

What is the one-dimensional Schrodinger equation? (D)

Answer 5

ih * ∂𝛹/∂t = - h^2 / 2m * ∂^2𝛹/dx^2 + V𝛹

Question 6

Quantum Theory

What is the one-dimensional stationary state Schrodinger equation? (D)

Answer 6

• h^2 / 2m * d^2𝜓/dx^2 + V𝜓 = E𝜓

Question 7

Quantum Theory

What are the stationary state solutions and associated energies of the one-dimensional particle in a box? (Q)

Answer 7

𝜓n(x)=Bsin(npix/a)

Energy: En=n^2 pi^2 h^2 / 2ma^2

Question 8

Quantum Theory

What are the wave functions of the one-dimensional particle in a box? (Q)

Answer 8

𝛹n(x,t)=Bsin(npix/a) * e^( -i n^2 pi^2 h t / 2 m a^2 )

Question 9

Quantum Theory

What is the ground state energy? (D)

Answer 9

When the possible energies of a quantum system are discrete and bounded below, the ground state energy is the smallest energy state

Question 10

Quantum Theory

What are the stationary state solutions and associated energies of the three-dimensional particle in a box? (Q)

Answer 10

𝜓n1,n2,n3(x,y,z)= Bsin(n1pix/a)sin(n2piy/b)sin(n3piz/c)

With energies En1,n2,n3
= pi^2 h^2 / 2m ( n1^2/a^2 + n2^2/b^2 + n3^2/c^2 )

Question 11

Quantum Theory

What does it mean to say an energy level E has d-fold degeneracy? (D)

Answer 11

It means the space of solutions to the stationary state schrodinger equation with energy level E has a dimension d>1. When d=1 we call E a non-degenerate energy state

Question 12

Quantum Theory

What is the correspondence principle? (D)

Answer 12

The tendency for quantum results to tend to the classical result as the quantum number tends to ∞ (as the energy of the system increases)

Question 13

Quantum Theory

What is the probability density function for the particles position? (D)

Answer 13

The square of the magnitude of the wave function

Question 14

Quantum Theory

What does it mean for a wave function to be normalised? (D)

Answer 14

For the integral over all space of the magnitude square of the wave function to be equal to 1

Question 15

Quantum Theory

What does it mean for a wave function to be normalisable? (D)

Answer 15

For the integral over all space of the magnitude square of the wave function to be strictly between 0 and ∞

Question 16

Quantum Theory

What is the continuity equation? (D)

Answer 16

∂ρ/∂t + ∇·j = 0

Question 17

Quantum Theory

What is the probability current j? (D)

Answer 17

j(X,t) = ih / 2m ( 𝛹*conj(∇𝛹)-conj(𝛹)∇𝛹 )

Question 18

Quantum Theory

What’s the rough proof of the continuity equation? (T)

Answer 18

Note ¦𝛹(x,t)¦^2 = conj(𝛹)𝛹.
Differentiate with product rule
Expand with Schrodinger
Then collect terms and note ∇^2 = ∇·∇

Question 19

Quantum Theory

What is the condition for int[R^3] ¦𝛹¦^2 dxdydz to be independent of t? (T)

Answer 19

j must tend to 0 faster than 1/¦x¦^2 as ¦x¦ ->∞

Question 20

Quantum Theory
Prove the theorem that if j tends to 0 faster than 1/¦x¦^2 as ¦x¦ -> ∞. Then the integral of R^3 of ¦𝛹¦^2 is independent of time. (T)

Answer 20

Idea of proof if to look at the derivative of the integral of a general space D with border S. Use continuity. Divergence. Then transform to spherical. Let S be sphere and show it tends to 0 as r->∞.
Proof in Chapter 3 notes

Question 21

Quantum Theory

3 conditions on solutions to the Schrodinger equation? (Q)

Answer 21

The wave function should be a continuous, single valued function. This means the probability density is single-valued with no discontinuities

The wave function should be normalisable

∇𝛹 should be continuous everywhere, except when there is an infinite discontinuity in V. This follows as discontinuity in ∇𝛹 => infinite discontinuity in ∇^2𝛹 => infinite discontinuity in V from Schrodinger

Question 22

Quantum Theory

When solving an IVP for a particle in a box, how to you calculate the coefficients cn? (Q)

Answer 22

[0]int[a] f(x)𝜓m(x) dx = [n=1]sum[∞] ( cn * [0]int[a] 𝜓n(x)𝜓m(x) dx ) = cm

Since [0]int[a] 𝜓m𝜓n dx = 0 as they are orthogonal
The 2nd equality comes from the fourier series for f(x)

Question 23

Quantum Theory

What is the probability of measuring the energy of the particle to be En? (Q)

Answer 23

¦cn¦^2

Question 24

Quantum Theory

Show [n=1]sum[∞] ¦cn¦^2 = 1 (T)

Answer 24

1
= [0]int[a] ¦𝛹¦^2 dx
= [m,n=1]sum[∞] of conj(cm)cn * e^-i(En-Em)t/h * [0]int[a] conj(𝜓m)𝜓n dx
= [n=1]sum[∞] ¦cn¦^2

2nd equality from series expansion of 𝛹. 3rd equality since 𝜓 is real