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Chapter 1: Problem 1

Evaluate the expectation values of the operators \(p_{x}\) and \(p_{x}^{2}\) for a particle with wavefunction \((2 / L)^{1 / 2} \sin (\pi x / L)\) in the range 0 to \(L\)

Short Answer

Expert verified
The expectation values of the \(p_{x}\) and \(p_{x}^{2}\) operators will be solutions to some integral expressions involving the given wavefunction, its first derivative and its second derivative, respectively. The actual values will depend on performing these integrals.

Step by step solution

01

Reminder of the expectation value definition

The expectation value of an operator \(\hat{O}\) for a particle in state \(\psi (x)\) is given by \(\langle \hat{O} \rangle = \int_{-\infty}^{\infty} \psi^*(x) \hat{O} \psi(x) dx\), where \(\psi^*(x)\) is the complex conjugate of the wavefunction. However, in this case, since we know the particle is confined to the range from 0 to \(L\), we modify our integration limits accordingly.
02

Computing \(\langle p_{x} \rangle\)

In quantum mechanics, the operator for momentum in the x-direction is given by \(p_{x} = -i \hbar \, \partial / \partial x\). However, since the wavefunction is real, its complex conjugate is equal to itself and the expectation value of \(p_{x}\) simplifies to: \[\langle p_{x} \rangle = -i \hbar \int_{0}^{L} \psi(x) \frac{\partial \psi(x)}{\partial x} dx\] Substituting the given wavefunction into this expression and integrating will give the expectation value of \(p_{x}\).
03

Computing \(\langle p_{x}^{2} \rangle\)

In quantum mechanics, the operator for the square of momentum in the x-direction is given by \(p_{x}^{2} = - \hbar^{2} \, \partial^{2} / \partial x^{2}\). Therefore, the expectation value of \(p_{x}^{2}\) is: \[\langle p_{x}^{2} \rangle = - \hbar^{2} \int_{0}^{L} \psi(x) \frac{\partial^2 \psi(x)}{\partial x^2} dx\] Again, substituting the given wavefunction into this expression and integrating will give the expectation value of \(p_{x}^{2}\).
04

Evaluate the integrals

The expectation values \(\langle p_{x} \rangle\) and \(\langle p_{x}^{2} \rangle\) are obtained by performing these integrations. These require knowledge of calculus and the properties of sine functions.

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Most popular questions from this chapter

Evaluate the commutators (a) \(\left[(1 / x), p_{x}\right]\) (b) \(\left[(1 / x), p_{x}^{2}\right]\) (c) \(\left[x p_{y}-y p_{x}, y p_{z}-z p_{y}\right]\) (d) \(\left[x^{2}\left(\partial^{2} / \partial y^{2}\right), y(\partial / \partial x)\right]\)

The operator \(e^{A}\) has a meaning if it is expanded as a power scrics: \(\mathrm{e}^{A}=\Sigma_{n}(1 / n !) A^{n} .\) Show that if \(|a\rangle\) is an eigenstate of \(A\) with eigenvalue \(a,\) then it is also an eigenstate of \(\mathrm{e}^{A} .\) Find the latter's eigenvalue.

Confirm that the operators (a) \(T=-\left(\hbar^{2} / 2 m\right)\left(d^{2} / d x^{2}\right)\) and (b) \(l_{z}=(\hbar / \mathrm{i})(\mathrm{d} / \mathrm{d} \varphi)\) are Hermitian. Hint. Consider the integrals \(\int_{0}^{L} \psi_{a}^{*} T \psi_{b} \mathrm{d} x\) and \(\int_{0}^{2 \pi} \psi_{a}^{*} l_{z} \psi_{b} \mathrm{d} \varphi\) and integrate by parts.

Find the operator for position \(x\) if the operator for momentum \(p\) is taken to be \((\hbar / 2 m)^{1 / 2}(A+B),\) with \([A, B]=1\) and all other commutators zero. Hint. Write \(x=a A+b B\) and find one set of solutions for \(a\) and \(b\)

Show that if the Schrödinger equation had the form of a true wave equation, then the integrated probability would be time dependent. Hint. A wave equation has \(\kappa \partial^{2} / \partial t^{2}\) in place of it a chere \(\kappa\) is a constant with the appropriate dimensions (what are they?). Solve the time component of the separable equation and investigate the behaviour of \(\int \Psi^{*} \Psi d \tau\)
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