物理代写|PHYS451 Quantum mechanics

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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 

Course Description
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）

Consider the following wave function:
$$
\psi(x, t)=A x e^{-\beta|x|+i \lambda t}
$$
where $\beta$ and $\lambda$ are some real constants and $\beta>0$.

(a) To determine the normalization factor, $A$, we need to ensure that the probability of finding the particle in all space is equal to $1$. That is, we need to evaluate the integral of $|\psi(x, t)|^2$ over all space and set it equal to $1$. We have: \begin{align*} \int_{-\infty}^{\infty} |\psi(x, t)|^2 dx &= \int_{-\infty}^{\infty} |Ax e^{-\beta |x| + i \lambda t}|^2 dx \ &= \int_{-\infty}^{\infty} A^2 x^2 e^{-2\beta |x|} dx \ &= 2A^2 \int_0^{\infty} x^2 e^{-2\beta x} dx \quad \text{(even function)} \ &= 2A^2 \frac{2}{(2\beta)^3} \ &= \frac{A^2}{\beta^3}. \end{align*} Setting this equal to $1$, we obtain:
$$
A=\beta^{3 / 2}
$$

(c) The standard deviation of $x$, $\sigma$, is given by: \begin{align*} \sigma &= \sqrt{\left\langle x^2\right\rangle – \langle x \rangle^2} \ &= \sqrt{\frac{3}{4\beta^2}} \ &= \frac{\sqrt{3}}{2\beta}. \end{align*}

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