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Chapter 1: Problem 10

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

Short Answer

Expert verified
The answers to the exercise are (a) \(-i\hbar/x^2\), (b) \(-2i\hbar / x^3\), (c) 0, and (d) \(2xy\).

Step by step solution

01

Evaluate Commutator (a)

The x operator in quantum physics represents multiplication by x and the momentum operator \(p_x = -i\hbar ( \partial / \partial x)\). Insert these into the commutator \([1/x, p_x]\), and simplify. The result will be \(-i\hbar/x^2\).
02

Evaluate Commutator (b)

In this case, we deal with the squared momentum operator. Substitute this value into the equation \([1/x, p_x^2]\). Next, expand and simplify the operation. The task becomes more difficult due to higher degree derivatives, but the method is the same. The result will be \(-2i\hbar / x^3\). This can be arrived at by using the result of commutator (a) and the commutation property.
03

Evaluate Commutator (c)

In this case, use the same values for the operators, but multiply them by constants. The constant does not influence the order of application of the operators, so they can be extracted outside the commutators. The commutator is zero in this case because of the commutation relations amongst different Cartesian components of position and momentum. The commutation of operators \( [p_{y}, p_{z}] = [p_x, p_z] = [p_x, p_y] = 0 \) and \([x, y] = [x, z] = [y, z] = 0\).
04

Evaluate Commutator (d)

Operator \(x^2\) represents \(x^2\) multiplication, and operator \(\partial^{2} / \partial y^{2}\) represents the second derivative with respect to \(y\). Operator \(y(\partial / \partial x)\) means taking the derivative with respect to \(x\) and then multiplying by \(y\). Put these into the equation \([x^2(\partial^{2} / \partial y^{2}), y(\partial /\partial x)]\), expand and simplify, this will yield the result of \( 2xy \)

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Most popular questions from this chapter

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 \(L\)

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.

Evaluate the commutator \(\left[l_{y}\left[l_{y}, l_{z}\right]\right]\) given that \(\left[l_{x}, l_{y}\right]=i \hbar l,\left[l_{y}, l_{z}\right]=i \hbar l_{x},\) and \(\left[l_{z}, l_{x}\right]=i \hbar l_{y}\)

Write the time-independent Schrödinger equations for (a) the hydrogen atom, (b) the helium atom, (c) the hydrogen molecule, (d) a free particle, (e) a particle subjected to a constant, uniform force. Hint. Identify the appropriate potential energy terms and express them as operators in the position representation.

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