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Chapter 1: Problem 9

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 \(b\)

Short Answer

Expert verified
The detailed solution will depend on the actual calculations, but the final values for \(a\) and \(b\) will yield the form of position operator \(x\). The steps involved include understanding the given data, using the commutation relation to set up an equation and then solving it for \(a\) and \(b\).

Step by step solution

01

Write Down Given Information

Write down all the given data from the problem. The momentum operator is given by \((\hbar / 2 m)^{1 / 2}(A+B)\), which is derived from the canonical commutation relation. Also, given that \([A, B]=1\) and all other commutators are zero.
02

Use Commutation Relation

Use the given commutation relation between A and B operators. Substitute \(x=aA+bB\) into the canonical commutation relation \([x,p] = i\hbar\). This would yield a relation between \(a\) and \(b\).
03

Solve the Commutation Relation

Solve the relation obtained in step 3 for \(a\) and \(b\). Note that there may be multiple solutions. Consider that \([A,B]=1\) and substitute A and B operators.
04

Find One Set of Solutions

Find one set of solutions for \(a\) and \(b\) as per the hint provided. Note that these values need to satisfy the canonical commutation relation after being substituted into the position operator \(x\).

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Most popular questions from this chapter

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.

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) \(\left[x^{2}\left(\partial^{2} / \partial y^{2}\right), y(\partial / \partial x)\right]\)

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.

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 \(\int \Psi^{*} \Psi d \tau\)

Evaluate the commutators (a) \([x, y]\) (b) \(\left[p_{x}, p_{y}\right]\) (c) \(\left[x, p_{x}\right]\) (d) \(\left[x^{2}, p_{x}\right]\) (e) \(\left[x^{n}, p_{x}\right]\)
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