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这是华盛顿州立大学量子力学课程的代写成功案例。
PHYS451课程简介
Prerequisites, PHYS 250 and PHYS 201 or PHYS 207. Students learn the basic principles of quantum mechanics, which describe atomic, molecular, and nuclear physics. Students learn general concepts, methods, analytic tools and advanced problem-solving skills. Topics include evidence for and origins of quantum mechanics, mathematical background, the postulates of quantum mechanics, one-dimensional systems, quantization of angular momentum, and three-dimensional quantum systems including the hydrogen atom. Letter grade with Pass/No Pass option. (Offered fall semester.) 3 credit
Prerequisites
In this course, students learn the basics of non-relativistic quantum mechanics. The course introduces the concept of the wave function, its interpretation, and covers the topics of potential wells, potential barriers, quantum harmonic oscillator, and the hydrogen atom. Next, a more formal approach to quantum mechanics is taken by introducing the postulates of quantum mechanics, quantum operators, Hilbert spaces, Heisenberg uncertainty principle, and time evolution. The course ends with topics covering the addition of angular momenta, spin, and some basic aspects of many-body quantum mechanics. The course will include two lectures per week accompanied by a recitation.
PHYS451 Quantum mechanics HELP（EXAM HELP， ONLINE TUTOR）
Verify the Heisenberg uncertainty principle for the following two states of one-dimensional harmonic oscillator:
(a) The ground state with the wave function $\psi_0(x)=\frac{\alpha^{1 / 4}}{\pi^{1 / 4}} e^{-\alpha x^2 / 2}$.
(b) The first excited state with the wave function $\psi_1(x)=\sqrt{2} \frac{\alpha^{3 / 4}}{\pi^{1 / 4}} x e^{-\alpha x^2 / 2}$.
In the above expressions $\alpha=\frac{m \omega}{\hbar}$, where $m$ is the particle mass and $\omega$ is the angular frequency of the oscillator.
Consider a particle in an infinite square box of length $a$. Assume that initially the particle is in the ground state. Then suddenly, at time $t=0^{+}$, the box is expanded instantaneously to the length of $2 a$.
(a) Find the coefficients of $\psi\left(x, t=0^{+}\right)$in the basis of the eigenstates of the new box (of length $2 a$ ).
(b) Will the system ever return to its initial state, and if so, at which time?
Consider the wavepacket:
$$
\psi(x)=A \exp \left[i k_0 x-\frac{\left(x-x_0\right)^2}{4 \sigma^2}\right]
$$
where $A, k_0, x_0$, and $\sigma$ are some real constants.
(a) Determine the normalization factor, $A$.
(b) Find the wave function in the momentum space, $\psi(k)$.
(c) Calculate $\langle x\rangle,\left\langle x^2\right\rangle$, and $\Delta x$.
(d) Calculate $\langle p\rangle,\left\langle p^2\right\rangle$, and $\Delta p$.
(e) Calculate the probability current, $j(x)$
Demonstrate that
$$
\delta(x)=\lim {\epsilon \rightarrow 0^{+}} \frac{1}{\pi} \frac{\epsilon}{x^2+\epsilon^2} $$ is a valid representation of the Dirac delta function. Namely, show that (a) $\int{-\infty}^{+\infty} \delta(x) f(x) d x=f(0)$ for any reasonably “nice” function $f(x)$.
(b) $\delta(x)=\delta(-x)$
(c) $x \delta(x)=0$.
(d) $\delta(c x)=\frac{1}{|c|} \delta(x)$
(e) $\delta^{\prime}(-x)=-\delta^{\prime}(x)$
(f) $x \delta^{\prime}(x)=-\delta(x)$
Consider a particle in an infinite square box of length $a$. Assume that initially the particle is in the ground state. Then suddenly, at time $t=0^{+}$, the box is expanded instantaneously to the length of $2 a$.
(a) Find the coefficients of $\psi\left(x, t=0^{+}\right)$in the basis of the eigenstates of the new box (of length $2 a$ ).
(b) Will the system ever return to its initial state, and if so, at which time?
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