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Chapter 1: Problem 3

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) \(\left[x^{n}, p_{x}\right]\)

Short Answer

Expert verified
The results of the commutators are: (a) 0, (b) 0, (c) \(i\hbar\), (d) \(2xi\hbar\), and (e) \(inx^{n-1}i\hbar = in\hbar x^{n-1}\)

Step by step solution

01

Evaluation of the first commutator

Since \(x\) and \(y\) are two different variables and thus independent, their commutator is simply zero. Hence, \([x, y] = 0\)
02

Evaluation of the second commutator

Again, since \(p_{x}\) and \(p_{y}\) are the momenta in two different, and thus independent, directions, their commutator is zero. Hence, \([p_{x}, p_{y}] = 0\)
03

Evaluation of the third commutator

It's a fundamental result in quantum mechanics that the commutator of the position and momentum in the same direction is equal to \(i\hbar\). Hence, \([x, p_{x}] = i\hbar\)
04

Evaluation of the fourth commutator

The operator \(x^{2}\) can be rewritten as \(xx\), which results in \([x^{2}, p_{x}] = [xx, p_{x}] = x[x, p_{x}] + [x, p_{x}]x = xi\hbar + i\hbar x = 2xi\hbar\)
05

Evaluation of the fifth commutator

To evaluate the commutator \([x^{n}, p_{x}]\), you need to iteratively apply the operator to \([x,x^{n-1}]\), using the relation worked out in step 4. After doing this, you will find \([x^{n}, p_{x}] = inx^{n-1}i\hbar = in\hbar x^{n-1}\)

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Most popular questions from this chapter

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 \(b\)

Are the linear combinations \(2 x-y-z, 2 y-x-z\) \(2 z-x-y\) linearly independent?

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.

Evaluate the commutators (a) \(\left[H, p_{x}\right]\) and (b) \([H, x]\) where \(H=p_{x}^{2} / 2 m+V(x) .\) Choose (i) \(V(x)=V,\) a constant, (ii) \(V(x)=\frac{1}{2} k_{t} x^{2},\) (iii) \(V(x) \rightarrow V(r)=e^{2} / 4 \pi \varepsilon_{0} r .\) Hint. For part (b), case (iii), use \(\left(\partial r^{-1} / \partial x\right)=-x / r^{3}\)

The ground-state wavefunction of a hydrogen atom has the form \(\psi(r)=N \mathrm{e}^{-b r}, b\) being a collection of fundamental constants with the magnitude \(1 / a_{0},\) with \(a_{0}=53 \mathrm{pm}\) Normalize this spherically symmetrical function. Hint. The volume element is \(\mathrm{d} \tau=\sin \theta \mathrm{d} \theta \mathrm{d} \varphi r^{2} \mathrm{d} r,\) with \(0 \leq \theta \leq \pi\) \(0 \leq \varphi \leq 2 \pi,\) and \(0 \leq r<\infty,\) 'Normalize' always means 'normalize to 1 ' in this text.
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