Wave mechanics and Quantum

Question 1

Describe what would happen classically in the photoelectric effect experiment.

Answer 1

Electrons should be emitted after some delay, with an energy dependent on the intensity of light.

Question 2

Describe what happens experimentally in the photoelectric effect experiment.

Answer 2

Electrons are always emitted instantaneously, even at low intensities, with energies independent of that intensity but dependent on the wavelength. No electrons are emitted below a certain wavelength.

Question 3

Plank equation for energy.

Answer 3

E=hf

Question 4

Describe what classically happens for spectral lines.

Answer 4

Electrons should not be stable in their orbits but should lose energy and spiral in towards the positively charged nucleus.

Question 5

Describe what experimentally happens for spectral lines.

Answer 5

The arrangement of electrons and nucleus is stable, with electrons occupying distinct energy levels, as shown by lines in spectra from discharge lamps.

Question 6

Equation linking angular momenta of electrons to quantised units.

Answer 6

mvr = lh(bar)

Question 7

Derive the energy level model for a H atom and the Rydberg.

Answer 7

Check derivation.

Question 8

de Broglie relation.

Answer 8

p = h(bar)k = h/λ

Question 9

§What is a wavepacket?

Answer 9

Combination of a range of wavelengths.

Question 10

From the idea of a wavepacket, simply derive a version of Heisenburg’s Uncertainty Principle.

Answer 10

Check derivation.

Question 11

Calculate the total energy of a hydrogen atom from KE and PE.

Answer 11

Check…answer should be a Rydberg.

Question 12

Show how measuring position and momentum of an electron accelerated through a well defined potential does not break the uncertainty principle.

Answer 12

Check derivation.

Question 13

Sum two travelling waves to make a standing wave.

Answer 13

Check

Question 14

Why can’t standing waves be used to describe the electron?

Answer 14

Antinodes mean places where electrons cannot exist.

Question 15

Why can’t a travelling wave be used to describe the electron?

Answer 15

Can’t exist over all space.

Question 16

What kind of wave functions do we use for electrons?

Answer 16

Complex.

Question 17

Born probability of a travelling wave function.

Answer 17

P(x,t) = |ψ(x,t)|^2 = ψ*.ψ = A^2

Question 18

Derive the time independent Schrödinger Equation in 1D

Answer 18

Check derivation.

Question 19

Under fixed boundary conditions, what happens when a travelling wave reaches the boundary?

Answer 19

It reflects and changes sign.

Question 20

Under free boundary conditions, what happens to a travelling wave when it reaches a boundary?

Answer 20

It reflects without a change in sign.

Question 21

Derive an expression for phase velocity.

Answer 21

Check

Question 22

State expression for group velocity.

Answer 22

Check

Question 23

Find both the phase and group velocities for a photon.

Answer 23

Check

Question 24

Find both the phase and group velocities for an electron.

Answer 24

Check.

Question 25

The first postulate of quantum mechanics.

Answer 25

For any particle moving in an external potential, there is an associated wave which is a single valued function of the spatial coordinates and of time. The wave function determines everything that can be known about the particle.

Question 26

The second postulate of quantum mechanics.

Answer 26

For every physical observable, Q (energy, momentum etc), there is an associated mathematical operator Qˆ. A measurement of the observable, Q, gives a result which is one of the eigenvalues of the eigenvalue equation for the operator. Qˆu(r) = qu(r) q is eigenvalue.

Question 27

The third postulate of quantum mechanics.

Answer 27

The functions u(r) are eigenfunctions of Q and form a complete orthonormal set so that any arbitrary normalised wave function ψ(r) can be expressed as a linear combination of the eigenfunctions.

Question 28

The fourth postulate of quantum mechanics.

Answer 28

If a particle has a wave function ψ(r), which may or may not be an eigenfunction of the observable Q, then the probability that a measurement of Q will result in the eigenvalue q is given by: P(q) = (∫ u*(r)ψ(r)dV)^2

Question 29

The fifth postulate of quantum mechanics.

Answer 29

The measurement of an eigenvalue, q, immediately changes the wave function of the particle from ψ(r) to u(r).

Question 30

The sixth postulate of quantum mechanics.

Answer 30

The mean value (aka expectation value) of the observable, Q, for particles with wavefunction, ψ(r), will = ∫ψ*(r)Qˆψ(r)dV

Question 31

State the momentum operator.

Answer 31

Check

Question 32

State the total energy operator.

Answer 32

Check

Question 33

State the time operator.

Answer 33

Check

Question 34

State the position operator.

Answer 34

Check

Question 35

State the potential energy operator.

Answer 35

Check.

Question 36

Derive the KE operator.

Answer 36

Check.

Question 37

Derive the angular momentum operator.

Answer 37

Check

Question 38

What happens when you use the energy operator on a free particle wave function.

Answer 38

Check The result is Plank's formula

Question 39

What happens when you use the momentum operator on a free particle?

Answer 39

Check The result is the de Broglie formula

Question 40

Derive the Heisenburg Uncertainty Principle using the commutation of operators.

Answer 40

Check derivation.

Question 41

Solve the time dependent Schrödinger equation

Answer 41

Check

Question 42

Solve the time independent Schrödinger equation

Answer 42

Check

Question 43

Find the total wavefunction putting together the time dependent solution and the spatial dependent solution.

Answer 43

Check.

Question 44

Normalise the wave function for a particle in a 1D box.

Answer 44

Check

Question 45

What is Hilbert space?

Answer 45

An infinite dimensional space that can be defined by eigenfunctions that form a complete orthonormal set (used as basis vectors).

Question 46

Even if the wavefunction is not an eigenfunction of the observable, what is the outcome always?

Answer 46

An eigenvalue.

Question 47

If a measurement is performed of any observable Q, and the eigenvalue q is obtained, what happens to the system?

Answer 47

The subsequent state of the system will be the eigenfunction u(r).

Question 48

What is non-locality?

Answer 48

Where a measurement at one location can affect the result at a subsequent without any signal passing between the two due to quantum entanglement.

Question 49

Derive the wavefunction for a particle of energy E incident on a potential step V.

Answer 49

Check derivation.

Question 50

Derive the wavefunction for a particle of energy E incident on a potential step V and V>E.

Answer 50

Check derivation.

Question 51

What is quantum tunnelling?

Answer 51

The phenomena where there is a finite probability of finding a particle in a forbidden region (potential barrier), and a finite probability that a particle will pass through a finite width barrier even though the potential is higher than the total energy.

Question 52

How does the probability of quantum tunnelling vary with width of the barrier?

Answer 52

Falls exponentially with increasing width.

Question 53

3 examples of quantum tunnelling.

Answer 53

STM Alpha decay Covalent bonding.

Question 54

How does quantum tunnelling work in STM?

Answer 54

The small electrical bias applied to the tip and the sample becoming a tunnelling gap.

Question 55

How does quantum tunnelling work in alpha decay?

Answer 55

An alpha particle must tunnel through a barrier formed by the short range attraction to other nucleons and the longer range electrostatic repulsion by the remaining protons.

Question 56

How does quantum tunnelling work in covalent bonding?

Answer 56

If the height of the potential barrier is not infinite, but greater than the total energy, then evanescent waves exist outside the well.

Question 57

State the Schödinger equation in 3D Cartesian coordinates.

Answer 57

Check

Question 58

State the Schödinger equation in 3D polar coordinates.

Answer 58

Check

Question 59

State the solution for the energy of a particle in a 3D box.

Answer 59

Check

Question 60

What is degeneracy?

Answer 60

When states with different quantum numbers have the same energy they are described as degenerate. If an energy state has n number of degenerate states, it has n-fold degeneracy.

Question 61

Show that x, y and z components of angular momentum do not commute.

Answer 61

Check

Question 62

Show that angular momentum squared and the z component of angular momentum commute.

Answer 62

Check

Question 63

Derive equations showing the spherical harmonics are eigenfunctions of the angular momentum squared operator and z component of the angular momentum operator.

Answer 63

Check derivation.

Question 64

What are the allowed values of the m quantum number?

Answer 64

-l ≤ m ≤ +l

Question 65

Solve the radial energy equation for the energy of a hydrogen atom.

Answer 65

Check

Question 66

Calculate the most probable radius of an electron in a hydrogen 1s orbital.

Answer 66

Check

Question 67

What is the eigenvalue of spin?

Answer 67

h(bar)^2s(s+1)

Question 68

What values of spin do electrons always take?

Answer 68

±1/2

Question 69

What values of spin do bosons always take?

Answer 69

Integer spins.

Question 70

State the Pauli exclusion principle.

Answer 70

No two indistinguishable fermions can occupy the same quantum state.

Question 71

Why do lower n valued orbitals tend to fill first?

Answer 71

Lower n values are more bound so fill first. For a given principle quantum number n, the larger l tend to have a larger mean radius and are more screened. This lifts the degeneracy such that lower l values are more bound than the high l vales within the same n shell so fill first.

Question 72

Hund's first rule.

Answer 72

The total spin S is the sum of the ms values for individual electrons. This is consistent with the Pauli exclusion principle, the electrons will fill to maximise S.

Question 73

Hund's second rule.

Answer 73

The total orbital angular momentum L, is the sum of the ml values for the individual electrons. This is consistent with the Pauli exclusion principle, the electrons will maximise L.

Question 74

Hund's third rule.

Answer 74

If the partially filled subshell is not more than half full, then the total angular momentum is h(bar)J where J = |L-S|, and if it is more than half full then J = L+S.

Question 75

Calculate the total angular momentum of a silicon atom and write down its term symbol.

Answer 75

Check

Question 76

For H2, using linear combinations of orbitals, show that cA = cB for bonding and cA = -cB for antibonding.

Answer 76

Check.