Principles of Quantum Mechanics

Question 1

What is the Hamiltonian?

Answer 1

The sum of the kinetic and potential energies of all particles associated with the system.

Question 2

What is the equation for wavenumber, k?

Answer 2

k = 2π / λ = 2πf / c

Question 3

State the time-dependent Schrodinger equation for a single particle of mass m in 1 dimension in a potential V (x).

Answer 3

(i ̄h) ∂Ψ / ∂t = - ( ̄h^2) / (2m) * (∂^2Ψ) / (∂x^2) + V(x) Ψ(x,t)

Question 4

State what it means for a wavefunction |φ〉to be an eigenfunction of an operator ˆA.

Answer 4

|φ〉 is an eigenfunction of an operator ˆA if:

ˆA|φ = λ |φ〉

where λ is the eigenvalue corresponding to the eigenfunction |φ〉.

Question 5

(State what it means for a wavefunction |φ〉to be an eigenfunction of an operator ˆA.)

Supposing the operator ˆA represents a physical quantity, what is the physical interpretation of the set of eigenvalues of ˆA?

Answer 5

The set of eigenvalues of ˆA corresponds to

the possible outcomes of a measurement

of the physical quantity represented by ˆA.

Question 6

Give the expression for the energy levels E(nx ny nz) of the 3-dimensional harmonic oscillator.

Answer 6

E(nx, ny, nz) = (nx + ny + nz + 3 / 2) ̄hω

Question 7

(Give the expression for the energy levels E(nx ny nz) of the 3-dimensional harmonic oscillator.)

What are the allows values of the quantum numbers nx, ny, nz?

Answer 7

Where:
nx = 0, 1, 2, …
ny = 0, 1, 2, …
nz = 0, 1, 2, …

Question 8

(Give the expression for the energy levels E(nx ny nz) of the 3-dimensional harmonic oscillator.)

(What are the allows values of the quantum numbers nx, ny, nz?)

Give an example of degeneracy for the 3-dimensional harmonic oscillator.

Answer 8

E(100) = E(010) = E(001) = 5 / 2 ̄hω

Question 9

What is the dirac notation?

Answer 9

〈φ|ψ〉 =∫ (∞, −∞) φ∗ψ dx

Question 10

How do you find the complex conjugate of something?

Answer 10

You find the complex conjugate by changing the sign of the imaginary part of the complex number.

Question 11

State the defining property of an even function f(x).

Answer 11

A function f (x) is called even when:

f (-x) = f (x)

Question 12

State the defining property of an odd function f(x).

Answer 12

A function f (x) is called odd when:

f (-x) = -f (x)

Question 13

Is the product of an even and an odd function even or odd?

Answer 13

The product of an even and an odd function is odd.

Question 14

Describe the quantum mechanical phenomenon of tunneling.

Answer 14

Tunneling is a quantum mechanical phenomenon

when a particle is able to penetrate through a potential energy barrier

that is higher in energy than the particle’s kinetic energy.

Question 15

Describe a technological application of tunneling.

Answer 15

A technological application of the phenomenon of tunneling is the Scanning Tunnelling Microscope (STM).

Here an atomically sharp tip is being moved across a surface.

The tunneling current of electrons (between the tip and the
surface) is measured and from this, the distance between the tip and the surface can be determined.

By scanning the surface, a height profile can be obtained at atomic resolution

Question 16

Express the z-component L(z) of the classical angular momentum (->)L in terms of the coordinates x, y, z and the momenta p(x), p(y), p(z).

Answer 16

L(z) = x p(y) - y p(x)

Question 17

Express the x-component L(x) of the classical angular momentum (->)L in terms of the coordinates x, y, z and the momenta p(x), p(y), p(z).

Answer 17

L(x) = y p(z) - z p(y)

Question 18

Express the y-component L(y) of the classical angular momentum (->)L in terms of the coordinates x, y, z and the momenta p(x), p(y), p(z).

Answer 18

L(y) = z p(x) - x p(z)

Question 19

What is the quantum-mechanical operator ^L(z)?

Answer 19

^L(z) = x ^p(y) - y^p(x)

Question 20

Give the equation for the Hamiltonian.

Answer 20

H^ = - (ħ^2 / 2m) (∂^2 / ∂x^2) + V(x)

Question 21

What is the operator form of the Schrodinger equation?

Answer 21

^H ψ = E ψ

Question 22

What is the equation for energy levels?

Answer 22

E(n) = (n + 1/2) ħ ω

Question 23

What is Born’s rule?

Answer 23

P(x) = |ψ(x, t)|^2

Question 24

What is a “good” quantum number?

Answer 24

If [ ˆA, ˆH ] = 0,

then eigenfunctions of ˆA at an initial point in time remain eigenfunctions for all time.

The corrresponding physical quantity A is conserved.

In this case, the quantum number associated with the operator
ˆA is said to be a good quantum number.

Question 25

What is the Heisenberg Uncertainty Principle?

Answer 25

∆x ∆p ≥ ħ / 2

Where:
∆x is the uncertainty in position
∆p is the uncertainty in momentum

Question 26

State the equation of motion of a classical harmonic oscillator with mass m and spring constant k.

Answer 26

m (..)x = F = - k x

ω = √(k / m)