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Chapter 1: Problem 13

Construct quantum mechanical operators in the position representation for the following observables: (a) kinetic energy in one and in three dimensions, (b) the inverse separation, \(1 / x,\) (c) electric dipole moment \(\left(\Sigma_{i} Q_{i} r_{i} \text { where } r_{i}\right.\) is the position of a charge \(Q_{i}\) ), (d) \(z\) -component of angular momentum \(\left(x p_{y}-y p_{x}\right),\) (e) the mean square deviations of the position and momentum of a particle from the mean values.

Short Answer

Expert verified
Kinetic Energy Operator for 1D and 3D: \(-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{dx^{2}}\) and \(-\frac{\hbar^{2}}{2 m}\left(\frac{d^{2}}{dx_{1}^{2}}+\frac{d^{2}}{dx_{2}^{2}}+\frac{d^{2}}{dx_{3}^{2}}\right)\), respectively. Inverse separation operator: \(1 / x\). Electric dipole moment operator: \(\Sigma_{i} Q_{i} r_{i}\). Z-component angular momentum operator: \(-i\hbar(x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x})\). Mean square deviation operator (position and momentum): \(\Delta^{2} x=x^{2} - \langle x \rangle^{2}\) and \(\Delta^{2} p=p^{2} - \langle p \rangle^{2}\), respectively.

Step by step solution

01

Constructing the kinetic energy operator

The kinetic energy operator in the position representation in one dimension is given by \(- \frac{\hbar^{2}}{2 m} \frac{d^{2}}{dx^{2}}\). If one is dealing with three dimensions then the kinetic energy operator becomes \(-\frac{\hbar^{2}}{2 m}\left(\frac{d^{2}}{dx_{1}^{2}}+\frac{d^{2}}{dx_{2}^{2}}+\frac{d^{2}}{dx_{3}^{2}}\right)\).
02

Constructing the inverse separation operator

The quantum mechanical operator in the position representation for inverse separation, \(1 / x\) simply remains \(1 / x\).
03

Constructing the electric dipole moment operator

The electric dipole moment operator in the position representation is given by \(\Sigma_{i} Q_{i} r_{i}\), similar to the classical expression.
04

Constructing the Z-component of angular momentum operator

Operator in position representation for z-component of angular momentum is given by \(x p_{y}-y p_{x} = -i\hbar(x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x})\).
05

Constructing the operator for mean square deviations

Observables corresponding to square of position and momentum deviation from mean value will be \(\Delta^{2} x=x^{2} - \langle x \rangle^{2}\) and \(\Delta^{2} p=p^{2} - \langle p \rangle^{2}\)

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Most popular questions from this chapter

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.

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.

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

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 \(b\)

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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