Question
It is desired to calculate the expectation value of the product of position $x$ and momentum $p_{x} .$ Show that neither$$\left\langle x p_{x}\right\rangle=\int_{-\infty}^{\infty} \Psi^{*} x\left(\frac{\hbar}{i} \frac{\partial}{\partial x}\right) \Psi d x$$ nor $$\left\langle x p_{x}\right\rangle=\int_{-\infty}^{-} \Psi^{*}\left(\frac{\hbar}{i} \frac{\partial}{\partial x}\right) x \Psi d x$$is acceptable since both lead to imaginary values. Show that$$\langle x p\rangle=\frac{1}{2} \int_{-\infty}^{\infty} \Psi^{*}\left[x\left(\frac{\hbar}{i} \frac{\partial}{\partial x}\right)+\left(\frac{\hbar}{i} \frac{\partial}{\partial x}\right) x\right] \Psi d x$$leads to a real value. Does $\langle x p\rangle=\langle x\rangle\langle p\rangle ?$ Hint: Integrate by parts.
Step 1
The expectation value of an operator $O$ in quantum mechanics is given by $\langle O \rangle = \int_{-\infty}^{\infty} \Psi^{*} O \Psi dx$, where $\Psi$ is the wave function and $\Psi^{*}$ is its complex conjugate. Show more…
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In calculating the expectation value of the product of position times momentum, an ambiguity arises because it is not apparent which of the two expressions $$ \begin{aligned} &\overline{x p}=\int_{-\infty}^{\infty} \Psi^{\prime} x\left(-i \hbar \frac{\partial}{\partial x}\right) \Psi d x \\ &\overline{p x}=\int_{-\infty}^{\infty} \Psi^{*}\left(-i \hbar \frac{\partial}{\partial x}\right) x \Psi d x \end{aligned} $$ should be used. (In the first expression $\partial / \partial x$ operates on $\Psi$; in the second it operates on $x \Psi$.) (a) Show that neither is acceptable because both violate the obvious requirement that $\overline{x p}$ should be real since it is measurable. (b) Then show that the expression $$ \overline{x p}=\int_{-\infty}^{\infty} \Psi^{*}\left[\frac{x\left(-i h \frac{\partial}{\partial x}\right)+\left(-i \hbar \frac{\partial}{\partial x}\right) x}{2}\right] \Psi d x $$ is acceptable because it does satisfy this requirement. (Hint: (i) A quantity is real if it equals its own complex conjugate. (ii) Try integrating by parts. (iii) In any realistic case the wave function will always vanish at $x=\pm \infty$.)
Schroedinger'S Theory Of Quantum Mechanics
Problems
Use the Schrödinger equation to show that the expectation value of the kinetic energy of a particle is given by $$ \left\langle E_{k}\right\rangle=\int_{-\infty}^{+\infty} \psi(x)\left(-\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi(x)}{d x^{2}}\right) d x $$
As shown in Example 7.2, the expectation value of a measurable quantity can be determined by 'sandwiching' the quantity in between ψ* and ψ and integrating over all space where ψ is nonzero: < x >= ∫[from a to 0] ψ* x ψ dx (1) (For real-valued ψ, such as sin(x), ψ = ψ*). Given a particle in a box spanning (0 < x < a) with ψ = q^2 a sin(nπx/a), determine the expectation value for the operators p̂ and p̂^2, where p̂ = -iℏ d/dx. p̂ is actually the momentum operator. Appendix B may provide help with the integrals.
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