Problem 3
Evaluate the commutator \(\left[d / d y, 1 / y^{2}\right]\) by applying the operators to an arbitrary function \(f(y)\).
Problem 4
Show a. \(\operatorname{that} \psi(x)=e^{-x^{2} / 2}\) is an eigenfunction of \(\hat{A}=x^{2}-\partial^{2} / \partial x^{2} ;\) and b. that \(\hat{B} \psi(x)\) (where \(\hat{B}=x-\partial / \partial x\) ) is another eigenfunction of \(\hat{A}\).
Problem 5
Another important uncertainty principle is encountered in time-dependent systems. It relates the lifetime of a state \(\Delta t\) with the measured spread in the photon energy \(\Delta E\) associated with the decay of this state to a stationary state of the system. "Derive" the relation \(\Delta E \Delta t \geq \hbar / 2\) in the following steps. a. Starting from \(E=p_{x}^{2} / 2 m\) and \(\Delta E=\left(d E / d p_{x}\right) \Delta p_{x}\) show that \(\Delta E=\mathrm{v}_{x} \Delta p_{x}\), b. Using \(v_{x}=\Delta x / \Delta t,\) show that \(\Delta E \Delta t=\Delta p_{x} \Delta x \geq \hbar / 2\). c. Estimate the width of a spectral line originating from the decay of a state of lifetime \(1.0 \times 10^{-9} \mathrm{s}\) and \(1.0 \times 10^{-11} \mathrm{s}\) in inverse seconds and inverse centimeters.
Problem 6
Evaluate the commutator \([x(\partial / \partial y), y(\partial / \partial x)]\) by applying the operators to an arbitrary function \(f(x, y)\).
Problem 7
Evaluate \([\hat{A}, \hat{B}]\) if \(\hat{A}=x+d / d x\) and \(\hat{B}=x-d / d x.\)
Problem 9
Evaluate the commutator \(\left[\hat{p}_{x}+\hat{p}_{x}^{2}, \hat{p}_{x}^{2}\right]\) by applying the operators to an arbitrary function \(f(x)\).
Problem 11
Evaluate the commutator \(\left[y^{2}, d^{2} / d y^{2}\right]\) by applying the operators to an arbitrary function \(f(y)\).
Problem 13
Evaluate the commutator \(\left[\left(d^{2} / d y^{2}\right) y\right]\) by applying the operators to an arbitrary function \(f(y)\).
Problem 14
If the wave function describing a system is not an eigenfunction of the operator \(\hat{B},\) measurements on identically prepared systems will give different results. The variance of this set of results is defined in error analysis as \(\sigma_{B}^{2}=\) \(\left\langle(B-\langle B\rangle)^{2}\right\rangle,\) where \(B\) is the value of the observable in a single measurement and \(\langle B\rangle\) is the average of all measurements. Using the definition of the average value from the quantum mechanical postulates, \(\langle A\rangle=\int \psi^{*}(x) \hat{A} \psi(x) d x\), show that \(\sigma_{B}^{2}=\left\langle B^{2}\right\rangle-\langle B\rangle^{2}\).
Problem 15
Apply the Heisenberg uncertainty principle to estimate the zero point energy for the particle in the box. a. First, justify the assumption that \(\Delta x \leq a\) and that, as a result, \(\Delta p \geq \hbar / 2 a .\) Justify the statement that, if \(\Delta p \geq 0\), we cannot know that \(E=p^{2} / 2 m\) is identically zero. b. Make this application more quantitative. Assume that \(\Delta x=0.35 a\) and \(\Delta p=0.35 p\) where \(p\) is the momentum in the lowest energy state. Calculate the total energy of this state based on these assumptions and compare your result with the ground-state energy for the particle in the box. c. Compare your estimates for \(\Delta p\) and \(\Delta x\) with the more rigorously derived uncertainties \(\sigma_{p}\) and \(\sigma_{x}\) of Equation (17.13).
