Register to reply

Wave function problem

by Gregg
Tags: function, wave
Share this thread:
Mar1-10, 12:57 PM
P: 463
1. The problem statement, all variables and given/known data
The function

[itex] \Psi(r) = A(2-{Zr\over a})e^-{Zr\over 2a} [/itex]

gives the form of the quantum mechanical wavefunction representing the electron
in a hydrogen-like atom of atomic number Z when the electron is in its first
allowed spherically symmetric excited state. Here r is the usual spherical polar
coordinate, but, because of the spherical symmetry, the coordinates θ and φ do
not appear explicitly in Ψ. Determine the value that A (assumed real) must have
if the wavefunction is to be correctly normalised, i.e. the volume integral of |Ψ|^2
over all space is equal to unity.

3. The attempt at a solution

[itex] {\int \int \int}_R |\Psi|^2 dV = 1[/itex]

[itex] \int _0^{\infty }\int _0^{2\pi }\int _0^{\pi }A^2e^{-\frac{Zr}{a}} \left(2-\frac{Zr}{a}\right)^2d\phi d\theta dr = 1[/itex]

Which implies

[itex] \int _0^{\infty }A^2e^{-\frac{\text{Zr}}{a}} \left(2-\frac{\text{Zr}}{a}\right)^2dr = {1\over 2\pi^2}[/itex]

This turns out to be

[itex] \frac{2aA^2}{Z} = \frac{1}{2\pi^2} [/itex]

[itex] A = \pm \frac{\sqrt{\frac{z}{a}}}{2\pi} [/itex]

This is wrong though?
Is the problem the fact that Psi(r) isnt a function of theta or phi?
Phys.Org News Partner Science news on
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
Mar1-10, 01:16 PM
HW Helper
P: 481
Looks to me you forgot the r2 sin term from the differential volume element (note that the way you did it, the volume of a unit ball would come out as 2pi2). See

Register to reply

Related Discussions
Hydrogen Atom Radial Wave Function (limit problem) Quantum Physics 7
Determine a function that calculates the nth weight factor of a wave function Advanced Physics Homework 0
Nonsinusodial wave function problem Introductory Physics Homework 2
Wave function proving problem Introductory Physics Homework 1
Particle in a box... wave function problem Advanced Physics Homework 8