University College London英国伦敦大学学院物理专业量子力学代考PHAS0042 Quantum Mechanics，UCL期末考试，开学补考欢迎来询，经验丰富！诸如：PHAS0041:-Solid State Physics固体物理学，PHAS0038 – Electromagnetic Theory电磁理论，MATH0025 Mathematics For General Relativity相对论，PHAS0025 – Mathematical Methods III数学方法这些课程都可以接！近两年针对UCL英国伦敦大学学院的exam代考考试经验丰富，通常国内下午5点开考，时长3-4小时，专业靠谱英国代考
The primitive vectors of the face-centred cubic (fcc)lattice are Not yet
a = (a/2)(O,1,1), b = (a/2)(1,0,1) and c = (a/2)(1,1,0) answered
so that points on the Bravais lattice have position vectors,
Rn1,n2,n3 = n1 a + n2 b +n3 c
The constant a is the lattice parameter and n1,n2 and n3 are integers. What is the shortest distance (in units of the lattice para
nneter a,otice have)? [15
the lattice and,for a given point, how many points surrounding it are this far away (ie. how many nearest neighbours does a poimarks]
- distance= 0.5 a,9 nearest neighbours
- distance = a, 12 nearest neighbours
- distance =0.7071 a, 12 nearest neighbours
- distance= 1a，8 nearest neighbours
- distance = 0.5a, 2 nearest neighbours
- None of these answers are correct
- distance = 0.7071a, 6 nearest neighbours
(a) Show that, by choosing suitable values for P and Q, Green’s Theorem in the
plane leads to the formula for the area enclosed by a loop i.e.,
xdy − ydx
(b) Use this to find the area of the closed curve defined by
x = cos
y = 3 sin
where 0 ≤ ≤ 2. [4 marks]
Consider the vector field, F = x2yz i+xy2z j+xyz2 k. Use the divergence theorem
” F · ds
over the surface of the unit cube defined by the ranges x = [0, 1]; y = [0, 1]; z = [0, 1].
Consider the vector field F = (2x+yz) i+(2y+xz) j+xyk. Using Stokes’ theorem,
show that ¸ F · dr = 0 around any closed curve. [2 marks]
Consider the vector field, F = y i+x k. Use Stokes’ theorem to find ¸ F·dr around
the circular loop in the xy-plane defined by x2 + y2 = a2. [4 marks]
For gravity, g, we can define the divergence as, ∇ · g = −4G, where G is the
gravitational constant and is the mass density. The divergence theorem states that,
g · ds = °V
∇ · g dV.
By applying the divergence theorem to a point within the Earth’s radius, a distance
r from the centre:
(a) Draw a sketch showing a suitable choice of surface, S. [2 marks]
(b) Show that the magnitude of g is given by:
g(r) = −
(Assume that the Earth has a constant density.) [2 marks]
The square OABC lies in the xy-plane and is defined by the points:
O = (0, 0, 0);A = (1, 0, 0);B = (1, 1, 0);C = (0, 1, 0). The vector field, b, is given by,
b = 2yz i + (x2 − y2) j + (y + x2 − z2) k.
(a) Evaluate the line integral, I1 = ¸ b · dr, following the path O-A-B-C-O.
(b) Determine ∇ × b. [2 marks]
(c) Evaluate, I2 = ˜OABC(∇ × b) · ds, over the square OABC. [4 marks]
Part I – Thermodynamics Question 4 (10 marks)
Consider an ideal Otto cycle, with a compression ratio of 11. At bottom dead centre before
the compression, the air is at 100 kPa, 300 K. The temperature at the end of the isentropic
expansion process is 700 K. Assume constant specific heats at room temperature, and the
following properties of air:
Cv = 0.718 kJ/kg.K, Cp = 1.005 kg/kg.K, R= 0.287 kJ/kg.K and γ = 1.4.
(a) Show and label the cycle on a P-v diagram, labelling all states, work and heat
transfer processes. (4 marks)
(b) Find the temperature and pressure at the beginning of the constant volume heat
addition process. (2 marks)
(c) Find the temperature and pressure at the beginning of the isentropic expansion
process. (2 marks)
(d) Determine the efficiency of the cycle. (2 marks)
Part II – Fluid Mechanics Question 3：
A cylindrical vessel is partially filled with water, and starting from rest the vessel is
rotated at a constant angular velocity 𝜔. The velocity 𝑉 within the fluid depends on
radial location 𝑟, the time from the start of rotation 𝜏, 𝜔 and the fluid properties.
(i) Find the non-dimensional groups that relate 𝑉 and the other parameters of
relevance. (8 marks)
(ii) If, in another experiment, honey is rotated in the same vessel at the same angular
speed, determine from your dimensionless parameters the ratio of the time for the honey and
water to attain the steady motion. Assume water and honey to be of the same density and
honey to be 2000 times more viscous than water. (4 marks)
(iii) At steady state, determine and explain briefly which non-dimensional groups will
be irrelevant. Then, from the remaining relevant group/s determine how 𝑉 will vary with 𝑟.
[6 pts] Calculate the product xp of the position and momentum uncertainties for the ground state
0 and for the rst excited state 1 of the harmonic oscillator. It may be useful to recall the denitions
of the position and momentum operators in terms of ^a and ^a+:
(^a + ^a+); ^p = i
(^a ^a+): (1)
2 What is Momentum Anyway?
[10 pts] Prove that
Hint: it is helpful to recall the following formulas:
@x2 + V (x) ; (2)
h^ Qi =
(x; t)^Q (x; t)dx; (3)
^p = ih
3 Double Delta Potential
[14 pts] Consider a particle of mass m in a potential given by
V (x) = [(x + a) + (x a)]; ; a > 0:
This system has two bound states for very large a, but only a single bound state for very small a.
- (2 pts) The stationary states in this potential may be taken to be either even or odd. Explain
why in one sentence. (Hint: either look at the form of V (x) or consider the parity operator.)
- (4 pts) Sketch the wave functions of the two bound states for very large a. Be sure to label
which of the two is the ground state and which is the excited state. Then sketch the wave
function of the single bound state for very small a.
For arbitrary a > 0, the bound state energies E are determined by the following transcendental
equation for the variable =
m 1; (x) even;
m ; (x) odd:
- (3 pts) Find the energies of the even and odd bound states in the limit as a goes to innity.
- (3 pts) Find the energy of the single bound state in the limit as a goes to zero.
- (2 pts) Estimate the value of a at which the system goes from having one bound states to two.
You are NOT expected to solve for a exactly.
4 Half Harmonic oscillator
[8 pts] Consider the half-harmonic oscillator potential (which represents, for example, a string which
can be stretched but not compressed), given by the potential
V (x) =
2m!2×2 for x > 0;
1 for x 0:
- (4 pts) What are the allowed energies? Explain your reasoning.
- (4 pts) What is the wave function for the ground state of the half-harmonic oscillator, written
as a function of x? Make sure your solution is normalized.
5 Two-Level System
[14 pts] Consider the photons and polarizers described in class, which is an instance of a two-level
system. Recall that the elements of the polarization vector correspond to probability amplitudes,
similar to the expansion coecients cn of wave functions.
- (3 pts) For a photon incident on a linear polarizer aligned in the ^x direction (meaning that it
transmits photons in the jxi-polarized state), what is the transmission probability if the initial
polarization vector of that photon is ~E = Ex^x + Ey ^y?
- (3 pts) A photon prepared in the jxi state is incident on a linear polarizer oriented at angle
relative to ^x. Solve for and sketch a graph of the probability of transmission (i.e. the probability
that the photon makes it through the polarizer) as a function of .
- (4 pts) As we showed in class, using the fjxi; jyig basis, an `x-polarizer’ transmits photons
polarized in the x-direction while re
ecting photons polarized in the y-direction. It makes a
measurement of the operator (expressed in the fjxi; jyig basis)
A polarizer rotated at +45 degrees relative to the ^x-axis measured the operator
What is the operator corresponding to a polarizer oriented at +30 degrees relative to the ^x-axis?
- (4 pts) Consider a particle prepared in the right-hand polarized state, j i = jRi = p1
(jxi + ijyi).
For times t < 0, the Hamiltonian for this particle is 0: in other words, it does not interact with
anything and nothing changes about its state. For times t 0, the following Hamiltonian is
; 0: (8)
Under the time evolution governed by ^H, is there ever a time t when the state becomes
[a] j (t)i = jLi = p1
(jxi ijyi)? Explain your reasoning. If yes, what is the rst time t
when this occurs?
[b] j (t)i = jxi? Explain your reasoning. If yes, what is the rst time t when this occurs?
6 Rigged Hilbert Space
[8 pts] Consider complex-valued functions f; g of a real variable x, with inner product dened as
1 f(x)g(x)dx. For the following functions, name the most restricted space to which they
belong: nuclear space, Hilbert space, extended space, or none of the above.
- (2 pts) f(x) = sin(x);
- (2 pts) f(x) = sin(x)ex2 ;
- (2 pts) f(x) = 1
- (2 pts) f(x) = ex2 .