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Chapter 1: Problem 7

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.

Short Answer

Expert verified
By using the normalization condition and breaking down the integrals, we find the normalization constant \(N = (1/a_{0})^{3/2}/(2\sqrt{\pi})\).

Step by step solution

01

Defining the wavefunction, volume element and normalization condition

The ground-state wavefunction for a hydrogen atom is given by: \[ \psi(r)=N e^{-br} \] Where \(N\) is the normalization constant we wish to find, \(b= 1/a_{0}\), and \(a_{0} = 53pm\). The volume element \(d\tau\) is defined in spherical coordinates as: \[d \tau=\sin \theta d \theta d \varphi r^{2} d r\] Where the limitations of the variables are \(0 \leq \theta \leq \pi\), \(0 \leq \varphi \leq 2 \pi\), and \(0 \leq r<\infty\), The normalization condition is given by: \[\int |\psi(r)|^2 d\tau = 1\] Which should hold for all choices of \(r\), \(\theta\), and \(\varphi\).
02

Substituting wavefunction into normalization condition

First, substitute the given wavefunction \(\psi(r)=Ne^{-br}\) into the normalization condition to set up the integral: \[\int |N e^{-br}|^2 r^{2} \sin \theta dr d\theta d\varphi = 1\] Since \(\psi(r)\) is real-valued, |N e^{-br}|^2 simplifies to \(N^2e^{-2br}\).\ Also, \(N^2\) is a constant, so can be pulled out of the integral: \[N^{2} \int_0^{2\pi} d\varphi \int_0^{\pi} \sin\theta d\theta \int_0^{\infty} e^{-2br} r^2 dr = 1\]
03

Solving the integrals

The three integrals can be solved separately because they are independent. The first two integrals over \(\varphi\) and \(\theta\) are trivial and lead to \(2\pi\) and \(2\) respectively.\ For the third integral, we can refer to a table of integrals or derived directly that \[\int_0^\infty x^n e^{-px} dx = \frac{n!}{p^{n+1}}\] For \(n=2\), \(p=2b\) and \(x=r\), this yields \(2/(4b^3)\) or \(1/(2b^3)\). Thus, the overall result of integration is: \[N^{2} * 2\pi * 2 * \frac{1}{2b^3} = 4\pi/b^3 = 1\]
04

Find the normalization constant \(N\)

Lastly, solve the above equation for the normalization constant \(N\). Taking the square root of both sides gives: \[N = \sqrt{1/(4\pi/b^3)}\] And solving for \(N\) gives: \[N = b^{3/2}/(2\sqrt{\pi}) = (1/a_{0})^{3/2}/(2\sqrt{\pi})\]

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Most popular questions from this chapter

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.

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.

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.

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.
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