?(a) Without evaluating any integrals, explain why you expect \(\langle x\rangle_{v}=0\)

Chapter 7, Problem P7E.17

(choose chapter or problem)

(a) Without evaluating any integrals, explain why you expect \(\langle x\rangle_{v}=0\) for all states of a harmonic oscillator. (b) Use a physical argument to explain why \(\left\langle p_{x}\right\rangle_{v}=0\). (c) Equation 7E.13c gives \(\left\langle E_{\mathrm{k}}\right\rangle_{v}=\frac{1}{2} E_{v}\). Recall that the kinetic energy is given by \(p^{2} / 2 m\) and hence find an expression for \(\left\langle p_{x}^{2}\right\rangle_{v}\). (d) Note from Topic 7C that the uncertainty in the position, \(\Delta x\), is given by \(\Delta x=\left(\left\langle x^{2}\right\rangle-\langle x\rangle^{2}\right)^{1 /2}\) and likewise for the momentum \(\Delta p_{x}=\left(\left\langle p_{x}^{2}\right\rangle-\left\langle p_{x}\right\rangle^{2}\right)^{1 / 2}\). Find expressions for \(\Delta x\) and \(\Delta p_{x}\) (the expression for \(\left\langle x^{2}\right\rangle_{v}\) is given in the text). (e) Hence find an expression for the product \(\Delta x \Delta p_{x}\) and show that the Heisenberg uncertainty principle is satisfied. (f) For which state is the product \(\Delta x \Delta p_{x}\) a minimum?

Text Transcription:

leftangle x rightangle _v=0

leftangle p_x rightangle_v=0

leftangle E_k rightangle_v=1/2 E_v

p^2 /2m

leftangle p_x^2 rightangle_v

Delta x

Deltax = (leftangle x^2 rightangle-x rightangle^2)^½

Deltap_x=(leftangle p_x^2 rightangle -p_x^2 rightangle)1/2

Delta_x

Delta p_x

leftangle x^2 rightangle_v

Delta x Delta p_x