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Chapter 1: Problem 8

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.

Short Answer

Expert verified
Both operators \(T\) and \(l_{z}\) are Hermitian. This is shown by demonstrating that the complex integral of the wavefunction \(\psi_{a}\), complex conjugated, multiplied by the result of the operator acting on \(\psi_{b}\) is equal to its complex conjugate.

Step by step solution

01

Apply Operator \(T\) and Integrate by parts

Let's consider two arbitrary wave-functions \(\psi_{a}\) and \(\psi_{b}\) and integrate the product of the first, complex conjugated, \(T\psi_{b}\) over the full length of \(L\). The integral becomes \(I_T=\int_{0}^{L} \psi_{a}^{*} T \psi_{b}dx = - \frac{\hbar^{2}}{2m} \int_{0}^{L} \psi_{a}^{*} \frac{d^{2}\psi_{b}}{dx^{2}} dx\). Then, according to the method of integration by parts, \(I_T = - \frac{\hbar^{2}}{2m} [\psi_{a}^{*}\frac{d\psi_{b}}{dx}]_{0}^{L} + \frac{\hbar^{2}}{2m} \int_{0}^{L} \frac{d\psi_{a}^{*}}{dx}\frac{d\psi_{b}}{dx} dx\).
02

Show Physical Necessity of Nulled Boundaries

For wave-functions, the necessity of them becoming null at their boundaries is based on physical grounds (the particle described by the wave-function shouldn't be found outside of these boundaries). Therefore, the first member of the right hand side of the previous expression disappears and we are left with: \(I_T = \frac{\hbar^{2}}{2m} \int_{0}^{L} \frac{d\psi_{a}^{*}}{dx}\frac{d\psi_{b}}{dx} dx\).
03

Repeat Steps 1 and 2 to Prove Both Operators are Hermitian

If the initial integral \(I_T\) was instead calculated with \(T\) acting on \(\psi_{a}^{*}\), the same expression would be obtained proving \(T\) to be a Hermitian operator. The same steps can be followed to prove \(l_{z}\) is also a Hermitian operator. Except now, the integration is performed from 0 to \(2\pi\) and \(\psi_a^*\) and \(\psi_b\) are functions of \(\varphi\) instead of \(x\).
04

Proving \(l_{z}\) is Hermitian

The initial integral with \(l_{z}\) acting on \(\psi_{b}\) is written as \(I_{l_{z}}=\int_{0}^{2 \pi} \psi_{a}^{*} l_{z} \psi_{b} d\varphi = \frac{\hbar}{i} \int_{0}^{2 \pi} \psi_{a}^{*} \frac{d\psi_{b}}{d\varphi} d\varphi\). Following the same reasoning as Step 2, and taking the complex conjugate of the result will yield the original integral - proving that the operator \(l_{z}\) is Hermitian.

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Most popular questions from this chapter

A particle is moving in a circle in the \(x y\) plane. The only coordinate of importance is the angle \(\varphi\) which can vary from 0 to \(2 \pi\) as the particle goes around the circle. We are interested in measurements of the angular momentum \(l_{z}\) of the particle. The angular momentum operator for such a system is given by \((\hbar / \mathrm{i}) \mathrm{d} / \mathrm{d} \varphi\) (a) Suppose that the state of the particle is described by the wavefunction \(\psi(\varphi)=N \mathrm{e}^{-\mathrm{i} \varphi}\) where \(N\) is the normalization constant. What values will we find when we measure the angular momentum of the particle? If more than one value is possible, what is the probability of obtaining each result? What is the expectation value of the angular momentum? (b) Now suppose that the state of the particle is described by the normalized wavefunction \(\psi(\varphi)=N\left\\{(3 / 4)^{1 / 2} \mathrm{e}^{-i \varphi}-(\mathrm{i} / 2) \mathrm{e}^{2 \text { ip }}\right\\} .\) When we measure the angular momentum of the particle, what values will we find? If more than one value is possible, what is the probability of obtaining each result? What is the expectation value of the angular momentum?

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

A particle in an infinite one-dimensional system was described by the wavefunction \(\psi(x)=\mathrm{Ne}^{-x^{2} / 2 r^{2}}\). Normalize this function. Calculate the probability of finding the particle in the range \(-\Gamma \leq x \leq \Gamma\). Hint. The integral encountered in the second part is the error function. It is available in mathematical software.

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.

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