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Modern Physics
Found in: Page 279
Modern Physics

Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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Short Answer

For the cubic 3D infinite well wave function

ψ(x,y,z)=AsinnxπxLsinnyπyLsinnzπzL show that the correct normalization constant is A=(2/L)3/2.

The normalization constant is 2L3/2.

See the step by step solution

Step by Step Solution

Step 1:  Given data

Cubic 3D infinite well wave function is given as:


Step 2:  Normalization condition

The normalization condition can be expressed as:


We know that the probability of finding the particle will always be 1.

Step 3:  To determine the normalization constant

Evaluate the normalization condition for a given 3D infinite well solution to determine the value of the normalization constant A as:

For a 3D infinite well, its wave function is given by



L is the edge length of the square well.

We know that the probability density ψx,y,z2 integrated over the volume of the 3D box must equal 1. Thus, we have




All three integrals multiplying each other are of the same form, for different values of nx,ny,nz so next evaluation just one of them for an arbitrary n.

Visualizing the dependence of the trigonometric functions graphically leads to the conclusion that,


Therefore, one segment can be solved as:


Thus, all three integrals multiplying each other in equation (1) are equal to each other and to L2.

Therefore, the equation (1) becomes,

A2L2L2L2=1 A2L23=1 A2=2L3 A2=2L3/2

Step 4:  Conclusion

Therefore, the normalization constant is 2L3/2.

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