Quantum Theory

Question 1

Give 5 problems which are unresolved without quantum theory

Answer 1

1. Photoelectric effect
2. Discrete emission/absorption spectra
3. Double slit experiment
4. De-Broglie’s matter-wave properties
5. Compton scattering

Question 2

Give the de Broglie relations

Answer 2

E = ℏω, p = ℏk, where E is the energy of a free particle, p is the momentum associated with a wave of angular frequency ω and k is the wave vector.

Question 3

State Schrodinger’s Equation

Answer 3

iℏ(∂Ψ/∂t) = −ℏ^2(∇^2)Ψ/2m + VΨ

Question 4

State the Stationary State Schrodinger Equation

Answer 4

−ℏ^2(∇^2)ψ/2m + Vψ = Eψ, where Ψ(x,t) = ψ(x)e^(−iEt/ℏ)

Question 5

For a particle with potential 0 inside the interval (0,a) and infinite potential outside, give the permitted energies of the particle.

Answer 5

E = En = (nπℏ)^2/2ma^2.

Question 6

Define the ground state

Answer 6

When the possible energies of a system are discrete and bounded below, the wavefunction with the lowest possible energy is called the ground state.

Question 7

For a particle with potential 0 inside the interval (0,a)x(0,b)x(0,c) and infinite potential outside, give the permitted energies of the particle.

Answer 7

E = E_(n1,n2,n3) = (πℏ)^2/2m[(n1/a)^2+(n2/b)^2+(n3/c)^2]

Question 8

Define degeneracy

Answer 8

The degeneracy of an energy level E of a system is the dimension of the solution space to the stationary state Schrödinger equation with energy E (though by convention, if this is one, it is said to be non-degenerate).

Question 9

Give the probability postulate

Answer 9

The function ρ(x, t) ≡ |Ψ(x, t)|^2 is a probability density function for the position of the particle, where Ψ(x, t) is the particle’s wave function.

Question 10

Define a normalisable wave function

Answer 10

A wave function Ψ is normalisable if 0 < triple integral over R3 of |Ψ(x, t)|^2 < ∞ for all t.

Question 11

Define a normalised wave function

Answer 11

A wave function Ψ is normalised if triple integral over R3 of |Ψ(x, t)|^2 =1 for all t.

Question 12

Define the correspondence principle

Answer 12

The tendency for quantum results to approach classical results for large quantum numbers.

Question 13

Define the expectation of a position function

Answer 13

The expectation value of a function of position f(x) is given by the triple integral over R3 of f(x)|Ψ(x, t)|^2

Question 14

Define the probability current of a wave function.

Answer 14

j(x,t) = iℏ/2m[Ψ(x,t){(∇Ψ)(x,t)}* − {Ψ(x,t)}*(∇Ψ)(x,t)].

Question 15

Give the conditions of a wave function

Answer 15

1. Wave functions must be continuous, single-valued functions. This condition ensures that the probability density is single-valued and has no discontinuities.
2. Wave functions must be normalizable.
3. ∇Ψ must be continuous everywhere, except where there is an infinite discontinuity in the potential V. This is necessary since a finite discontinuity in ∇Ψ implies an infinite discontinuity in ΔΨ, and thus from the Schrodinger equation an infinite discontinuity in V.

Question 16

Define the potential of the one-dimensional harmonic oscillator

Answer 16

The one-dimensional harmonic oscillator potential for angular frequency ω is V(x) = 1/2m(ωx)^2.

Question 17

Define an even parity state

Answer 17

In one dimension a stationary state wave function satisfying ψ(−x) = ψ(x) is said to describe even parity state.

Question 18

Define an odd parity state

Answer 18

In one dimension a stationary state wave function satisfying ψ(−x) = -ψ(x) is said to describe odd parity state.

Question 19

State the allowed energies of a one-dimensional harmonic oscillator

Answer 19

The energies of the one-dimensional quantum harmonic oscillator of angular frequency ω are E = En = (n + ½)ℏω.

Question 20

Define the complex inner product of a wave function

Answer 20

Let ϕ and ψ be wavefunctions. Then we define the complex inner product ⟨ϕ|ψ⟩ as the integral from -∞ to ∞ of ϕbar(x)ψ(x).

Question 21

Define a Hilbert space

Answer 21

A complete complex inner product space is called a Hilbert space.

Question 22

State the Hilbert space postulate

Answer 22

The states of a quantum system are elements of a Hilbert space.

Question 23

Define adjointness

Answer 23

If A is a linear operator acting on a complex inner product space, then the adjoint A∗ of A satisfies ⟨A∗ϕ|ψ⟩ = ⟨ϕ|Aψ⟩, for all ϕ, ψ ∈ H.

Question 24

Define self-adjointness

Answer 24

A linear operator A is self-adjoint if A∗ = A.

Question 25

State the observables postulate

Answer 25

The observables of a quantum system are given by self-adjoint linear operators on the space of states.

Question 26

Define the position operator

Answer 26

The position operator X in three dimensions has components Xi that are defined on wave functions ψ ∈ H via (Xiψ)(x) = xiψ(x). In vector form, Xψ = xψ.

Question 27

Define the momentum operator

Answer 27

The momentum operator P has components Pi that are defined on differentiable wave functions ψ by (Piψ)(x) = −iℏ∂ψ/∂(xi)(x). In vector form, Pψ = −iℏ∇ψ.

Question 28

Define the Hamiltonian operator

Answer 28

The Hamiltonian operator for a particle of mass m moving in a potential V is H = |P|^2/2m + V(X).

Question 29

Give the Hamiltonian version of the Schrodinger equation

Answer 29

The Schrodinger equation for time-dependent states ψ(t) is iℏ∂ψ(t)/∂t = Hψ(t), with stationary state ψ(t) = ψe^(−iEt/ℏ) of energy E satisfies Hψ = Eψ.

Question 30

Define an eigenstate

Answer 30

Given an observable A, then a state ψ ∈ H satisfying Aψ = αψ for some constant α ∈ C, is called an eigenstate of A, with eigenvalue α.

Question 31

Define a spectrum

Answer 31

The spectrum of an observable A is the set of all eigenvalues of A.

Question 32

State the Dirac span postulate

Answer 32

An observable A in quantum theory is required to have a complete set of eigenstates. That is, there is a set of orthonormal eigenstates {ψn} of A, such that any ψ ∈ H can be written as a linear combination of ψn.

Question 33

State the quantum measurement postulate

Answer 33

The possible outcomes of a measurement of an observable A are given by the eigenvalues {αn} of A. If the system is in a normalized quantum state ψ, then the probability of obtaining the value αn in a measurement, for a non-degenerate eigenvalue αn, is Pψ(obtaining αn) = |⟨ψn|ψ⟩|2.

Question 34

State the collapse of the wave function postulate

Answer 34

If one measures the observable A and obtains the non-degenerate eigenvalue αn, then immediately after the measurement the quantum state of the system is the eigenstate ψn ∈ H with Aψn = αnψn.

Question 35

Define the expectation of an operator

Answer 35

The expectation value of an operator A in a normalized quantum state ψ is Eψ(A) ≡ ⟨ψ|Aψ⟩.

Question 36

Define the identity operator

Answer 36

The identity operator 1 satisfies 1ψ = ψ, for all ψ ∈ H

Question 37

Define a non-negative operator

Answer 37

An operator A is said to be non-negative if ⟨ψ|Aψ⟩ ≥ 0, for all ψ ∈ H.

Question 38

Define the dispersion of an operator

Answer 38

The dispersion of an observable A in a normalized quantum state ψ is ∆ψ(A) ≡ sqrt(Eψ((A − Eψ(A)1)^2) = sqrt(Eψ(A^2) − (Eψ(A))^2).

Question 39

Define the commutator of two operators

Answer 39

The commutator of two operators A and B is [A, B] ≡ AB − BA.

Question 40

Give the canonical commutations

Answer 40

[Xi , Pj] = iℏδij1
[Xi, Xj] = 0 = [Pi , Pj].

Question 41

Give Leibniz’s rule of commutations

Answer 41

[A, BC] = B[A, C] + [A, B]C

Question 42

Give Jacobi’s identity of commutations

Answer 42

[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0.

Question 43

State Heisenberg’s uncertainty principle

Answer 43

For normalized ψ we have ∆ψ(X)∆ψ(P) ≥ 1/2ℏ and equality holds if and only if ψ(x) = exp[(x − µ)^2/2s0ℏ + γ], for some negative constant s0, and complex constants µ, γ ∈ C.

Question 44

Define a minimum uncertainty state

Answer 44

A wave function ψ such that ∆ψ(X)∆ψ(P) = 1/2ℏ.

Question 45

Define the raising/lowering operators

Answer 45

a± = (∓iP + mωX)/sqrt(2mωℏ)

Question 46

Define the number operator

Answer 46

N = (a+)(a-)

Question 47

Define the orbital angular momentum operator

Answer 47

The angular momentum operator L = (X2P3 − X3P2, X3P1 − X1P3, X1P2 − X2P1).

Question 48

Define the Levi-Cevita alternating symbol

Answer 48

The Levi-Civita alternating symbol ϵijk, defined by ϵijk = +1 if ijk is an even permutation of 123, −1 if ijk is an odd permutation of 123 and 0 otherwise.

Question 49

Define the angular momentum operator

Answer 49

Any self-adjoint vector operator J, with components Ji, i = 1, 2, 3, satisfying [Ji , Jj ] = iℏ times the sum by k of ϵijkJk.is called an angular momentum operator.

Question 50

Define total angular momentum

Answer 50

The total angular momentum is defined by J^2 = (J1)^2 + (J2)^2 + (J3)^2.

Question 51

Define the raising and lowering operators of angular momentum

Answer 51

J± = J1 ± iJ2.

Question 52

Give the Pauli matrices

Answer 52

The 2x2 matrices σ1 = (0 1), (1 0), σ2 = (0 −i) (i 0), σ3 = (1 0) (0 −1).

Question 53

Give the spin representation of angular momentum

Answer 53

Ji = ℏσi/2, where σi are the Pauli matrices are the matrix representation of spin.

Question 54

Give Coulemb’s potential

Answer 54

V(r) = e1e2/(4πrϵ0)

Question 55

Give the Laplacian in spherical coordinates

Answer 55

∇^2 = (1/r^2)d/dr[(r^2)d/dr] + (1/(r^2)sinθ)d/dθ[(sinθ)d/dθ] + (1/(r^2)sin^2(θ))d2/dϕ^2

Question 56

Give the Bohr radius

Answer 56

The Bohr radius is a = 4πϵ0ℏ^2/me^2.