Chapter 9 The Heisenberg Uncertainty Relations

9.1 Introduction

9.2 Derivation of the Uncertainty Relations

9.3 The Minimum Uncertainty Packet

9.4 Applications of the Uncertainty Principle

Exercise 9.4.1 Consider the oscillator in the state $|n=1\rangle$ and verify that

\left\langle\frac{1}{X^2}\right\rangle \simeq \frac{1}{\left\langle X^2\right\rangle} \simeq \frac{m \omega}{\hbar}

Exercise 9.4.2 (1) By referring to the table of integrals in Appendix A.2, verify that

\psi=\frac{1}{\left(\pi a_0^3\right)^{1 / 2}} e^{-r / a_0}, \quad r=\left(x^2+y^2+z^2\right)^{1 / 2}

is a normalized wave function (of the ground state of hydrogen). Note that in three dimensions the normalization condition is

\begin{aligned} \langle\psi \mid \psi\rangle & =\int \psi^*(r, \theta, \phi) \psi(r, \theta, \phi) r^2 \mathrm{d} r \mathrm{d}(\cos \theta) \mathrm{d} \phi \\ & =4 \pi \int \psi^*(r) \psi(r) r^2 \mathrm{d} r=1 \end{aligned}

for a function of just $r$ .

(2) Calculate $(\Delta X)^2$ in this state [argue that $(\Delta X)^2=\dfrac{1}{3}\left\langle r^2\right\rangle$ ] and regain the result quoted in Eq. (9.4.9).

(3) Show that $\langle 1 / r\rangle \simeq 1 /\langle r\rangle \simeq m e^2 / \hbar^2$ in this state.

Exercise 9.4.3 Ignore the fact that the hydrogen atom is a three-dimensional system and pretend that

H=\frac{P^2}{2 m}-\frac{e^2}{\left(R^2\right)^{1 / 2}} \quad\left(P^2=P_x^2+P_y^2+P_z^2, R^2=X^2+Y^2+Z^2\right)

corresponds to a one-dimensional problem. Assuming

\Delta P \cdot \Delta R \geqslant \hbar / 2

estimate the ground-state energy.

Exercise 9.4.4 Compute $\Delta T \cdot \Delta X$ , where $T=P^2/2m$ . Why is this relation not so famous?

目录

Chapter 9 The Heisenberg Uncertainty Relations

9.1 Introduction

9.2 Derivation of the Uncertainty Relations

9.3 The Minimum Uncertainty Packet

9.4 Applications of the Uncertainty Principle

9.5 The Energy-Time Unvertainty Relation