MY-ASSIGNMENTEXPERT™可以为您提供sydney PHYS3034 Quantum mechanics量子力学课程的代写代考和辅导服务！
这是悉尼大学量子力学课程的代写成功案例。
PHYS3034课程简介
Quantum statistical physics has revolutionized the world we live in- providing a profound understanding of the microscopic world and driving the technological revolution of the last few decades. Modern physics increasingly relies on solving equations using computational techniques, for modelling anything from the big bang to quantum dot lasers. Building on 2000-level physics, this unit will develop the full formalism for deriving properties of individual atoms and large collections of atoms, and introduce advanced numerical techniques. You will start from Schroedinger’s equation and derive the full properties of hydrogen atoms, and systems of particles. You will study perturbation techniques qualitatively, including for the interaction of radiation with atoms. You will study the theoretical foundation of statistical mechanics, including both classical and quantum distributions. You will apply a variety of numerical schemes for solving ordinary and partial differential equations, learn about the suitability of particular methods to particular problems, and their accuracy and stability. The module includes computational lab sessions, in which you will actively solve a range of physics problems. In completing this unit you will gain understanding of the foundations of modern physics and develop skills that will enable you to numerically solve complex problems in physics and beyond.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. demonstrate an understanding of key concepts in two foundation areas of physics – quantum mechanics of atoms and statistical physics
- LO2. apply these concepts to develop models, and to solve qualitative and quantitative problems in scientific contexts, using appropriate mathematical and computing techniques as necessary – Compare and critique different models in quantum and statistical physics
- LO3. design computer programs to solve physical problems
- LO4. compare and critique different approaches to numerically solving physical problems
- LO5. communicate scientific information appropriately, through written work
- LO6. analyse a physical problem in quantum physics and statistical physics and develop a formalism appropriate for solving it
- LO7. demonstrate a sense of responsibility, ethical behaviour, and independence as a learner and as a scientist.
PHYS3034 Quantum mechanics HELP（EXAM HELP， ONLINE TUTOR）
Normalization. Check that if $\Psi\left(r_1 \cdots r_n\right)$ is a normalized and (anti)symmetric wavefunction on $n$ particles, then
$$
|\Psi\rangle \equiv \sum_{r_1 \cdots r_n} \Psi\left(r_1 \cdots r_n\right)\left|r_1 \cdots r_n\right\rangle
$$
is normalized, $\langle\Psi \mid \Psi\rangle=1$.
(Interpret the sum over $r$ as an integral if you like.)
Multiple photons on paths of an interferometer.
One way to make a qbit is out of the two states of a photon moving on the upper and lower paths of an interferometer. On such a qbit, a half-silvered mirror $\mathbf{H}$ acts as a unitary gate, as indicated at left. (The dot below the mirror specifies a sign convention, to be explained below.)
On the other hand, photons are bosons. This means that if
$\mathbf{a}^{\dagger}|0,0\rangle \equiv|1,0\rangle$ is a state with one photon on the upper path
of the interferometer, then
$\frac{\left(\mathbf{a}^{\dagger}\right)^n}{\sqrt{n !}}|0,0\rangle \equiv|n, 0\rangle$ is a state with $n$ photons on the upper path.
Similarly, define
$\frac{\left(\mathbf{b}^{\dagger}\right)^n}{\sqrt{n !}}|0,0\rangle \equiv|0, n\rangle$ to be a state with $n$ photons on the lower path
of the interferometer. (Note that $[\mathbf{a}, \mathbf{b}]=0=\left[\mathbf{a}, \mathbf{b}^{\dagger}\right]$ – they are independent modes.)
Now suppose we direct these two paths through a half-silvered mirror, as in the figure. A half-silvered mirror acts as a Hadamard gate
$$
\mathbf{H} \equiv \frac{1}{\sqrt{2}}\left(\boldsymbol{\sigma}^x+\boldsymbol{\sigma}^z\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
1 & 1 \
1 & -1
\end{array}\right)
$$
on the qbit made from the one-photon states. (The dot tells us where to put the negative entry.)
Some warm-up questions:
(a) What is the state $|0,0\rangle$ ? How does $\mathbf{H}$ act on $|0,0\rangle$ ?
(b) How does $\mathbf{H}$ act on $|2,0\rangle$ and $|0,2\rangle$ ?
(c) How does $\mathbf{H}$ act on the operators $\mathbf{a}^{\dagger}$ and $\mathbf{b}^{\dagger}$ (in order that the above relations are realized)?
Brain-warmer: a beam of particles. Suppose the occupation numbers for a state of bosons satisfy
$$
n_{\vec{p}}=c e^{-\alpha\left(\vec{p}-\vec{p}0\right)^2 / 2} . $$ (a) Determine $c=c\left(n, \alpha, p_0\right)$ so that the average density is $$ n=\int \mathrm{d}^3 p n{\vec{p}} \text {. }
$$
(b) Check that with this normalization, in the thermodynamic limit of $N \rightarrow \infty$ at fixed $n=N / V$, the pair correlation function is
$$
g(x-y)=1+e^{-(x-y)^2 / \alpha} .
$$
Further evidence for the clumping tendencies of bosons.
Consider again the model of a $1 \mathrm{~d}$ crystalline solid that we discussed in class: It consists of $N$ point masses, coupled to their neighbors:
$$
\mathbf{H}0=\sum{n=1}^N\left(\frac{\mathbf{p}^2}{2 m}+\frac{1}{2} \kappa\left(\mathbf{q}n-\mathbf{q}{n-1}\right)^2\right)=\sum_{{k}} \hbar \omega_k\left(\mathbf{a}k^{\dagger} \mathbf{a}_k+\frac{1}{2}\right) . $$ Assume periodic boundary conditions $\mathbf{q}_n=\mathbf{q}{n+N}$, so that the allowed wavenumbers are
$$
{k} \equiv\left{k_j=\frac{2 \pi}{N a} j, \quad j=1,2 \ldots N\right} .
$$
Consider a state with two phonons defined by
$$
\left|k_1, k_2\right\rangle \equiv \mathbf{a}{k_1}^{\dagger} \mathbf{a}{k_2}^{\dagger}|0\rangle .
$$
(a) In the state $\left|k_1, k_2\right\rangle$, what is the probability of finding two phonons at the location $x_1$ ?
Do this problem both using a first-quantized point of view and using the algebra of creation and annihilation operators. Make sure your answers agree!
[Warning: the statement of this problem is deceptively simple.]
(b) Make sure your probabilities add up to one.
(c) Compare your result to the answer that would obtain if the particles were distinguishable (and occupied the same two single-particle states). Do bosons clump?
(d) Does the story change if $k_1=k_2$ ?
(e) Bonus problem: for two fermions in the state $\left|k_1, k_2\right\rangle$, what is the probability of finding one at $x_1$ and one at $x_2$ ? Check that your probabilities add to one. (It is possible to do this part in parallel with the others.)
MY-ASSIGNMENTEXPERT™可以为您提供SYDNEY PHYS3034 QUANTUM MECHANICS量子力学课程的代写代考和辅导服务！
