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Chapter 3: 3.10P (page 106)

Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its momentum? If not, why not?

Short Answer

Expert verified

The wavefunction ψnis not a momentum eigenfunction.

The magnitude of the momentum is a constant, and it is given by,

|p|=2mE=nπℏa

Step by step solution

01

Concept used

The wave function of the infinite square well's stationary states is calculated as follows:

ψn(x)=2asin(nπxa)

Where a is the width of the well.

02

Calculate the momentum

The wave function of the infinite square well's stationary states is calculated as follows:

p^=-iℏddx

We can check that directly as:

p^ψn=-iℏddx2asinnπxa=-iℏ2anπacosnπxa=-iℏnπacotnπxaψn(x)

Since the momentum operator doesn't yield the original wave function multiplied by a constant, then the wavefunction ψnis not an eigenfunction of momentum. The mean momentum is:

⟨p⟩=∫0aψnp^ψndx=-i2nπℏa2∫0asinnπxacosnπxadx=0

Which means the particle is just as likely to be found traveling to the left as to the right. The magnitude of momentum is a constant that is given by,

|p|=2mE=nπℏa

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