Quantum Questions

Question 1

Why can we use the results derived for the hydrogen atom to work out the orbital angular momentum, z component of orbital angular momentum and the orbital magnetic moment for any atom, but not for the energy?

Answer 1

The radial component has a Z dependence, the angular parts do not.

Question 2

Explain any limitations of the Schrodinger equation

Answer 2

Does not include spin orbit effects and so fails to predict sodium doublet and other phenomena due to energy level splitting

Question 3

Explain any limitations of the bohr model

Answer 3

Guesses quantisation, breaks heisenbergs uncertainty principle as it assumes fixed orbits and fixed trajectories, its derived assuming no uncertainty in the electrons position and velocity. Does not include any spin. Does not have any indication of degeneracy of each energy level.

Question 4

What are the Conditions for a quantum Harmonic Oscillator to be in its classical limit?

Answer 4

E»(Hbar)*w

Question 5

List the differences between Classical and Quantum Harmonic Oscillators

Answer 5

Quantum has a Zero Point energy, classically the lowest energy state is zero
Quantum energy states are discrete, classical is continuos
PDF’s for classical systems have maximums at either side, a ground state has a maximum at the centre of the potential

Question 6

Why can you not define mj unless you have an external magnetic field?

Answer 6

mj represents the total angular momentums projection onto the Z axis. We need an external B field to define the Z axis, else due to spherical symmetry as the Z axis is not well defined, neither is mj.

Question 7

Explain what |ψ|^2 is a measure of

Answer 7

|ψ|^2 Is a measure of the particles PDF. It is integrated to give the probability of finding a particle within a certain region.

Question 8

Why it is incorrect to say P(r)dr = |ψ|^2dr and why the correct expression is P(r)dr = |ψ|^2 dV where P(r) is the probability density function of the particle.

Answer 8

|ψ|^2 dr returns the probability of finding the particle at some distance r away from the origin WITHOUT taking into account the whole shell of points at that distance.
|ψ|^2 dV returns the same probability, but factoring in the shell of points which the particle could be at if it was at a distance r away.

Question 9

How many radial zeros does a wave function have with quantum numbers n and l

Answer 9

n-l-1

Question 10

How do we measure the energy of the electronic states?

Answer 10

Spectroscopy - Look at the light emitted from electrons transitions between particular states

Question 11

What is the balmer series

Answer 11

One of the 6 named sets hydrogen spectral emission lines. Describes any transition where the final state is n = 2 - All light in this series is visible.

Question 12

Describe what determines whether an emission line in the emission spectrum of hydrogen will be strong or weak.

Answer 12

Strong if changes in the following quantum numbers are:
L = +-1
J= +- 1, 0
mj=+-1 , 0

Question 13

Explain what natural or lifetime broadening is and the physical reason for the natural relaxation of excited electrons.

Answer 13

Natural or lifetime broadening:
This is the most fundamental mechanism and derives from the uncertainty principle that links the
uncertainty in energy ∆E_N (N for natural broadening) with the uncertainty in time ∆t,
∆E_N ∆t ∼ h bar
• A perfectly defined energy level for which ∆E = 0 must have an infinite lifetime.
• This may be true of the ground state, but not the excited states where given time they relax
back to the ground state, hence ∆t < ∞ and so ∆E > 0.

Physical reason:
Short lived Particle Antiparticle pairs constantly being created which can perturb our excited electron causing it to de excite.

Question 14

Derive an expression for the total broadening of a transition and show it is equal to:
hbar(1/ tau_ f + 1/tau _i) where tau is the average lifetime of an excited state, f is the final state and i is the initial state.

Answer 14

Start with the HUP:
delta E * tau = h bar
delta E _ transition = delta E _f + delta E _ i

Question 15

Explain Pressure or collision broadening and how it can be minimised.

Answer 15

The time between collisions of gas atoms can be shorter than the natural lifetime. This can cause
premature relaxation and emission of a photon. This reduces the lifetime and hence increases the
uncertainty ∆E_C in the energy.

At higher pressures the time τ_C between collision will be shorter than at lower pressures. (C for collision)

Minimise P to minimise delta E.

Question 16

Explain doppler broadening and how it can be minimised

Answer 16

If a transition of an atom that is stationary with respect to a photo detector has frequency f0
then atoms travelling towards (+) and away from (−) a detector with a velocity v will have an
apparent Doppler modified emission frequency.

Minimise T to minimise delta E

Question 17

If quantum states of the hydrogen atom have a φ dependence of the form exp(iαφ) where α is a
constant, explain why α must be an integer (i.e., the quantum number ml)

Answer 17

Need exp(iα(φ+2pi))=exp(iαφ) as φ +2pi returns us back to the same point in space, the wave function must be single valued and so must have the same value at this point and so the above equation must be satisfied. It is only satisfied if α is an integer as sin / cos ( α (x + 2pi) ) only returns the same value i

Question 18

Why cant you measure all 3 components of L at the same time?

Answer 18

Fundamental QM- measurement collapses the wavefunction to an eigen state with the measured value Eg measure the Lz component of a wave function. Wave function collapses to one of its eigen states with the measured value of Lz. This eigen state is not an eigen function of the Lx operator or the Ly operator and so measuring Lx or Ly subsequently will change the eigen state to a different one, which will mean we have lost all knowledge about what Lz is now.

Question 19

What happens when take an isolated electron and apply a weak magnetic field along the Z axis?

Answer 19

The magnetic moment due to spin of the electron will try to align with the B field but will fail due to the quantisation of Sz. There will be a net torque on the system as such and the electron will begin to precess about the Z axis.

Question 20

Evaluate L dot S to show its equal to 1/2 (J - L - S)^1/2

Answer 20

Start from j^2=(L^2+S^2)^2 ,expand RHS and factorise.

Question 21

Describe the important result of the first order time independent pertubation theory derivation.

Answer 21

We are allowed to use the ψ of the uncoupled Schrodinger equation
to compute the small
change in energy of a state if we perturb the system in some way. That is, we can use the
solutions we know from the ideal system to work out the energies of a non-ideal (perturbed)
system.

Question 22

What is the Zeeman effect and how does it affect electron energy levels? How does the size of this effect compare with the SO energy splitting?

Answer 22

A weak external magnetic field causes the energy of a state to be perturbed by different amounts based on the the mj value of the state.
As the external B field is weak compared to the internal B field, the E splitting due to the SO effect must be greater than the E splitting due to the Zeeman effect.

Question 23

Explain why putting neutrons through a B field will be deflected.

Answer 23

Neutrons they have magnetic moments that are thus affected differently by an external magnetic field, based on what direction the spin magnetic moment points in. Deflection of neutrons through a B field proves the existence of spin magnetic moments.

Question 24

List the assumptions made when considering the electronic states of many electron atoms.

Answer 24

• Electrons feel only 2 forces:
Attraction to the nucleus
Repulsion from other electrons.

• The time independent Schr¨odinger equation cannot be solved explicitly.

• Electrons feel the average effect of repulsion from
other electrons, leading to an effective potential energy for each electron that is still a
function of the radial distance r only.

Question 25

What is the main effect of repulsion between electrons on a given electron in a many electron atom?

Answer 25

Screening. The electron will feel a lower effective nuclear charge due to the other electrons.

Question 26

How does Zeff vary with:
n?
l, for a given n?
a probability density mostly outside the charge sphere of all the electrons?
other electrons

Answer 26

Zeff decreases as n increases
Zeff larger for smaller l
Zeff tends to 1.

Question 27

Explain why for many electron atoms it is possible that the 4s state is at a lower energy than the 3d state?

Answer 27

The S orbitals have the largest sized “bump” in their PDF that occurs close to the nucleus for a given n.
This little bump can drop the energy s shell for that level below the energy of some of the
sub-shells in a lower n state.

Question 28

Explain the structure of the periodic table in terms of the quantum numbers n and l

Answer 28

the different blocks of the periodic table correspond to the different l , (s p d and f ).
Can work out the electron configuration by finding position left to right and then up to down till you arrive at an elements position. See https://www.youtube.com/watch?v=Aoi4j8es4gQ&t=325s at 6:00 for example.

Question 29

Which electrons are involved in chemical reactions and optical spectra?

Answer 29

Electrons in partially filled shells are called valence electrons and are the ones involved in
chemical reactivity, the formation of bonds, and in optical spectra.

Question 30

Explain why there is a doublet in the emission spectra of sodium

Answer 30

The 3p state is split due to spin orbit effects and so its transition to the 3s state will give 2 possible photons closely spaced in their energies.

Question 31

Why are some excited states “Long lived”

Answer 31

They are forbidden to decay back to the ground state by selection rules

Question 32

Explain how a He Ne Laser works

Answer 32

He atoms are electrically excited into meta stable states where they can not relax from (violates selection rules) these excited He atoms instead transfer their energy to Ne atoms which have an energy state at a similar level to that of the excited He atoms. Neon decays to a lower energy state and gives of a photon.

Question 33

Explain what the different quantum numbers represent

n l ml and ms represent.

Answer 33

n - principal quantum number represents energy of orbital.
l- orbital angular momentum - gives shape of orbital
ml - magnetic quantum number - gives specific orbital amongst a specific set of of the orbitals for a given l.
ms - spin quantum number - tells you whether the system is in spin up or spin down.

Question 34

What does hunds rule state

Answer 34

Every orbital in a sublevel is singly occupied before any orbital is doubly occupied due to spin

Question 35

What is some experimental evidence which supports the particle like nature of light

Answer 35

Compton Scattering and photo electric effect

Question 36

What is a wave function?

Answer 36

Ψ describes the evolution of the probability distribution

of a quantum system.

Question 37

How does normalisation change the energy level associated with an eigen function

Answer 37

It doesn’t

Question 38

What are the necessary conditions the Schrodinger equation must satisfy

Answer 38

Conservation of energy, linearity in wave function, solutions in free space are sinusoidal,

Question 39

Properties of the wave function

Answer 39

Finite, Continuous, Single valued, continuous in its first differential .

Question 40

Why cant n = 0 for the infinite square well

Answer 40

psi = 0 => particle has no probability of existing anywhere. Not true.

Question 41

What is one explanation of zero point energy

Answer 41

The HUP, if particle had no energy when in ground state, delta P = 0, so delta x = inf, which it is not, so delta P must be finite as well.

Question 42

Why must electrons have spin

Answer 42

Due to the pauli exclusion principle

Question 43

Why are the energy levels for electrons negative

Answer 43

As they are bound states

Question 44

Why is the decay of an electron to a lower energy state a homogeneous poisson process

Answer 44

As each decay is an independent even with a mean time of tau

Question 45

Define spin for an electron

Answer 45

its the intrinsic angular momentum of the electron

Question 46

What does Degeneracy determine for a particular n state

Answer 46

Determines the maximum possible number of electrons in any given n, energy level (2n^2)

Question 47

What is the Copenhagen interpretation of QM

Answer 47

A wavefunction is made up of a superposition of ‘x’ possible values, once you make a measurement the wavefunction collapses onto one of its components with the probability of being in that state given by the square of the weighting factors (c).

Question 48

What frequency of light do you need to shine on an electron precessing about the z- axis due to an external magnetic field to cause it to flip spin states (ie from spin up to spin down)

Answer 48

Same frequency as the electron rotating about the z axis