T (K) T (K)

The data in each graph has been fitted with two linear equations over the temperature ranges shown in the table. Each equation has the form

c_p = A + B.T

where A and B are constants with units ]K~^xg~^r and ]K~²g~^r respectively. The break between the two equations is shown by the dashed line at the given temperature.

[3+(3+3+3+6+2) = 20 marks]

Hydrogen gas (H₂) is being used as an alternative fuel for vehicles. An attractive feature of hydrogen is that when it burned in the vehicle engine it produces only water vapour as exhaust by the chemical reaction:

2%(g) + O2S) - 2顷3)

Using the data in this table, give answers to the following questions.

Note: 1 atm = atmospheric pressure = 101 kPa

(i) Provide an estimate of the energy produced by burning 1 mol of H₂ gas with this reaction. State any assumptions you make.

(ii) The usual way to make hydrogen is by the "steam-methane-reforming process⁷⁷ using methane (CH₄) produced by natural gas wells. The reaction is:

CM) + 加。(9)- 3H2(g) + CO(g)

By considering a system of C, H and 0 atoms that is subject to an interaction with the external environment that causes the system to undergo this process, determine the enthalpy to produce 1 mol of H₂. State the assumptions you make and provide a physical explanation for the sign of your answer.

(iii) A cleaner way of making hydrogen is by electrolysis of liquid water using electricity generated by renewable or nuclear power. Calculate the enthalpy to make 1 mol of H₂ by this process. Compared to (ii), which process would you consider to be more efficient?

(iv) The final step after the H₂ is produced is to compress the gas into the fuel tank of a vehicle. Calculate how much work must be done to perform an adiabatic compression of 5 kg of H₂ at atmospheric pressure into a car fuel tank with a volume of 125 I. Be sure to provide a physical interpretation of the sign of your answer. This is sufficient H₂ gas to drive the car 400 km under normal circumstances.

(v) After the adiabatic compression, the gas will be very hot so in practice it is allowed to cool before being transferred into the car. Write down an expression for the ratio

of the entropy before compression to after compression assuming the gas from the factory and the gas transferred to the tank in the car are both at 298 K. This expression differs a lot from 1. Provide a qualitative explanation for the reasons for the change.

(b) The following diagram illustrates five different microstates (labelled A - E) of a system

> of 10 magnetic dipoles of paramagnetic spin-1/2 particles in an external magnetic field B

A： TTTTTTTlll

B： TTTTTTTTTT

C： TTTTTTTTll

D: TTUUUU

E-. TTTTUiiii

(i) In terms of the magnetic moment of a single dipole,卩,what is the energy of each microstate?

(ii) Which microstate belongs to the macrostate with the highest probability? Provide a physical argument to justify your answer.

(iii) Which microstate belongs to the macrostate with the highest entropy?

(iv) Does microstate B represent a thermal equilibrium state of the system at some temperature T? Use the second law of thermodynamics to explain your answer.

[(2+4+3+7+4)+(2+1+1+2) = 26 marks]

Part B: Classical Physics

Question 3 [29 marks]

(a) A mass, m, hanging from a spring, of length I with spring constant k under gravity g. is set in motion so the mass remains in the x — y plane at all times with the motion comprising a superposition of swing and oscillations. The mass of the spring is negligible compared to the mass m.

y — axis I %

(i) By introducing two variables: the angle 9 and the extension of the spring, r, from its equilibrium length I (see diagram), show that the Cartesian coordinates of the mass are:

% = (Z + r) sin 0

y = (Z + r) cos 0

(ii) Write down an expression for the total kinetic energy, T, of the system as a function of (r, 0).

(iii) Write down an expression for the potential energy, V, of the system taking the zero of gravitational potential energy to be at the anchor point of the spring.

(iv) Using your results, determine the Lagrangian for this system as a function of (r_f 0).

(v) Identify any ignorable coordinates and determine an expression for any associated constants of the motion. What is the physical interpretation of any constants of the motion you have identified?

(vi) Define the Hamiltonian, H, for this system in terms of the Lagrangian. Is H a constant of the motion? Justify your answer by showing that H is the total energy of the system.

(Question 3 continues on next page)

(b) The transverse oscillation mode of a tri-atomic linear molecule is very important for the absorption and re-radiation of energy in our atmosphere which is part of the Greenhouse effect. As a first approximation to model the transverse oscillation mode, consider a simplified linear molecule of three atoms of equal mass m that oscillates as in the diagram where the atom x-coordinates_, = I, x₂ = 0 (numbering the atoms left to right)

remain fixed and only the y-coordinates change so that y_r = y₃.at all times. You can also approximate the potential well in which the atoms oscillate to have an energy that depends only on the separation of the atoms, r (see diagram below-left), according to V = -/cr^z where k is a constant.

(i) Show that the atom separation is:

^r = J" + (% +，2

(ii) Write down the Lagrangian of the system as a function of the y-coordinates of the atoms.

(iii) Hence determine Lagrange⁵ s equations for the equations of motion.

(iv) From Lagrange⁵ s equations and the method of trial solutions, or otherwise, find the non-trivial oscillation frequency of the system in terms of k and m. Be sure to state any assumptions you make.

[(2+3+1+1 +4+4)+(2+3+3+6) = 29 marks]

Part C: Quantum Physics

(a) In a photoelectric effect experiment what particular interaction requires a quantum mechanical description? And, briefly explain two characteristics of the photoelectric effect that require this quantum mechanical interpretation.

(b) In a photoelectric effect experiment the work function of a particular metal is 0 = 5.3 eV. Find a value for the longest wavelength of the illuminating light for photo-excited electrons to be ejected from this metal surface?

(c) Find a value for the de Broglie wavelength of a 5.3 eV electron.

(d) In the tunnelling section we discussed an alternative to obtain electron emission from a metal surface where a field emission (tunnelling) process is used.

(i) For a 5.3 eV electron emitted from a metal surface via a field emission process, what is the kinetic energy of the electron at the point it exits the tunnel barrier, i.e. at the point it exits the classically forbidden region? Explain briefly.

(ii) In broad terms, what feature of the probability amplitude for the wavefunction in the tunnel barrier makes the tunnelling probability particularly sensitive to the width of the barrier?

(e) For a normalised wavefunction,寸(％) , that is an eigenfunction of the kinetic energy operator, [K] . Show by explicit calculation of AK = y/(K²) — (K)² that the kinetic energy, K , is sharp?

(f) For the ID infinite square-well potential, we found in lectures that the uncertainty in the x- component of momentum is given by:

nnh

x = ~r

where L is the width of the potential well. We also found that the uncertainty in the position of a particle in the ID infinite well is given by:

For the ground state, show whether or not these results are consistent with the uncertainty principle.

[3+2+3+(2+1 )+5+3= 19 marks]

Question 5 [20 marks]

Consider the ID finite asymmetric triangular potential well shown in the figure below where:

(

oo for x < 0

-L/q+毕％ for 0 < x < L

0 for x > L

As per lectures, we can use the time-independent formalism to extract useful information about the spatial variation of the wavefunction with respect to the spatial variation in the potential. For region III: x > L and a particle with energy E₂ > 0 ：

(a) Write down the time-independent Schrodinger equation (TISE) that applies in region III.

(b) Show that = Ce^lk，x + De~^lk，x , where C and!) are constants, is a suitable general solution to the TISE for region III and find an expression for the angular wavenumber, k\ in terms of the energy, E₂ , and other parameters.

For region III: x > L and a particle with energy < 0 :

(c) Show that your trial general solution for i/j_}}} in part (b) is still a suitable solution to the TISE for region III but that now k^r is complex. Hence show that the wavefunction can be rewritten as 仇〃 =C^l e~^ax + D^!e^ax and find an expression for a in terms of the energy,

and other constants.

(d) By applying the boundary condition that pertains as x oo modify your general wavefunction from part (c) to provide a wavefunction that meets this condition.

For region II: 0 < % < L and a particle with energy < 0 :

(e) Write down the time-independent Schrodinger equation (TISE) that applies in region II.

(f) By making appropriate substitutions (which you don't have to do here so don't panic) it is possible to simplify the TISE for this region to the following form:

d²ib

-—T — wxb = 0 dw^z

Solutions to this equation are Airy functions, which are combinations of Bessel functions and modified Bessel functions. There are two general types of Airy functions, examples of which are shown in the following figure:

1.00

0.75

0.50

0.25

0.00

-0.25

-0.50

(source: Wikipedia)

Briefly explain whether you expect the Airy functions of the first kind, AiQx) , or of the second kind, Bi(x) , to be appropriate solutions for this asymmetric triangular potential well in the region Q < x < L and < 0 and briefly explain the reason for your choice.

(g) On your exam answers page make a sketch of the ID finite asymmetric triangular potential well and add a rough sketch of what you expect the ground state wavefunction to look like in regions I and II i.e. for a particle with energy E_r < 0. Annotate your plot to indicate significant features.

(h) Finally, on your exam answers page add a rough sketch of what you expect the wavefunction to look like in regions I, II, and III for a particle with energy 監 > 0 (unbounded). Annotate your plot to indicate significant features.

[1+4+3+2+1 +1 +4+4 = 20 marks]

Question 6 [13 marks]

Consider the ID potential step shown in the figure below and consider individual electrons being incident on the potential step from the left:

Using our plane wave representation for the electron, a general wavefunction that is a suitable solution to the TISE that pertains to Region I is:

pi(x) = Ae^ikx + Be~^ikx

where A and B are constants and:

and, a general wavefunction that is a suitable solution to the TISE that pertains to Region II is: = Ce^ik，x + De~^ik，x

where C and D are constants and:

k^r =

(a) For an electron incident on the potential step from the left briefly explain what condition we use to set D = 0 for the wavefunction that applies in Region II.

(b) Write down the set of equations that result from applying the boundary conditions that

pertain at % = 0.

(c) The equations from part (b) can be used to find the reflection coefficient which is given by:

B ² (k —k')2

R =— =

A Q + R)2

Based on this expression for R write down a suitable expression for the transmission coefficient, T .

(d) For an electron of energy E = 2U₀ :

(i) Show that k^r = V2 k

(ii) and, hence find a value for the percentage of electrons that would be reflected under this condition.

(e) How does the quantum mechanical result differ from the classical expectation?

[2+2+2+(3+3)+1 = 13 marks]

(a) For the following wavefunction:

寸(0,(/)) = A sin² 9e—2i6

where A is a normalisation constant:

(i) Show whether or not the wavefunction is an eigenfunction of [L_z] , the angular momentum projection operator, and if it is then find a value for the magnetic quantum number, m_t .

(ii) Show whether or not the wavefunction is an eigenfunction of [L²] and if it is then find a value for the orbital angular momentum quantum number, I .

... .... 1 15

(iii) Show that the wavefunction is normalised if A =- —.

^{v 7} 4p27T

Potentially useful information:

一 f 2 1

sin⁵ x dx = — cos x + — cos³ % — - cos⁵ x + C

J 3 5

(b) Given that for H-like atoms wavefunctions that are suitable solutions to the relevant TISE with the same n but differing I and m_t are degenerate, which, if any, of the following superpositions will also be solutions of the TISE? Briefly explain your reasoning.

屮 I =沂(寸2,0,0 + Wi,o,o)

V2

0〃 =扬(02,1,1 +02,1,-1)

^III =插(02,1,1 + 02,1,0)

0W =、(02,0,0 + 02,1,0)

0V =扬(04,2,2 +02,1,1)

[(4+10+5) + 4 = 23 marks]