B.6 Chapters 10 to 12

B.6 Chapters 10 to 12

Problem 1:

Consider a quantum system describing a free particle in one spatial dimension, of size $L$ (the wavefunction satisfies $\psi (q,t)=\psi (q+L,t),$ ). If the wavefunction at time $t=0$ is given by

$$\psi (q,0)=C\left(\sin \left(\frac{6\pi }{L}q\right)+\cos \left(\frac{4\pi }{L}q+{\varphi }_0\right)\right)$$

where $C$ is a constant and ${\varphi }_0$ is an angle, find the wavefunction for all t. For what values of $C$ is this a normalized wavefunction $\begin{array}{r}(\int |\psi (q,t){|}^{2}dq=1)?\end{array}$

Problem 2:

Consider a state at $t=0$ of the one dimensional free particle quantum system given by a Gaussian peaked at $q=0$

$$\psi (q,0)=\sqrt{\frac{C}{\pi }}{e}^{-C{q}^2}$$

where $C$ is a real positive constant.

Show that the wavefunction $\psi (q,t)$ for $t>0$ remains a Gaussian, but one with an increasing width.

Now consider the case of an initial state $\psi (q,0)$ with Fourier transform peaked at $k=k_0$

$$\tilde{\psi }(k,0)=\sqrt{\frac{C}{\pi }}{e}^{-C(k-k_0{)}^2}$$

What is the initial wavefunction $\psi (q,0)?$

Show that at later times $|\psi (q,t){|}^2$ is peaked about a point that moves with velocity $\frac{\hslash k_0}{m}$

Problem 3:

Show that the limit as $T\to 0$ of the propagator

$$U(T,q_T-q_0)=\sqrt{\frac{m}{i2\pi T}}{e}^{-\frac{m}{i2T}(q_T-q_0{)}^2}$$

is a $\delta$-function distribution.

Problem 4:

Use the Cauchy integral formula method of section 12.6 to derive equation 12.9 for the propagator from equation 12.13.

Problem 5:

In chapter 10 we described the quantum system of a free non-relativistic particle of mass $m$ in ${\mathbf{R}}^3$ . Using tensor products, how would you describe a system of two identical such particles? Find the Hamiltonian and momentum operators. Find a basis for the energy and momentum eigenstates for such a system, first under the assumption that the particles are bosons, then under the assumption that the particles are fermions.

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

来源版本：2025-10-20

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