Eigenfunctions versus wavefunctions

[resurgance2001]

Hi - hope that someone can help me with this.

I am new to quantum mechanics - trying to answer a question about eigenfunctions and don't have a decent textbook at the moment.

Can someone tell me please, what is the difference between a wavefunction and an eigenfunction for a particle in an infinite square well?

Cheers

Peter

 

[pmb_phy]

Hi - hope that someone can help me with this.

I am new to quantum mechanics - trying to answer a question about eigenfunctions and don't have a decent textbook at the moment.

Can someone tell me please, what is the difference between a wavefunction and an eigenfunction for a particle in an infinite square well?

Cheers

Peter

An eigen function is the $$\psi$$ in the relationship A$$\psi$$ = a$$\psi$$. the $$\psi$$ can be either a wave function or can represent the state of a system, i.e. $$\psi$$ can be a "ket", which is a quantity (a generalized vector) which represents the system. A wavefunction, in general, can be either an eigenfunction/eigenket or a linear sum of eigenfunctions/eigenkets.

Pete

 

[Rach3]

There is a particular subset of wavefunctions known as "eigenfunctions" of an observable A, they are defined as those satisfying, for some scalar a,

$$\hat{A}\psi(x)=\alpha \psi(x)$$

(This is linear algebra actually.)

 

[Eye_in_the_Sky]

As you may know, the eigenvalues which one customarily computes in connection with the infinite square-well are energy eigenvalues: E₀, E₁, E₂, ...

Corresponding to each of these energy eigenvalues is a wavefunction φ₀(x),
φ₁(x), φ₂(x), ...

These particular wavefunctions, φ_n(x), are said to be the energy eigenfunctions associated with the infinite square-well. Thus, for example, a particle in the state φ_n(x) will have energy E_n.

But the state of a particle in the well doesn't have to be just a particular one of these φ_n(x). The state could be any normalized complex-valued function ψ(x) whose value is zero for x outside of the well. Such a ψ(x) is said to be a wavefunction for a particle in an infinite square-well.

Thus, every energy eigenfunction is a wavefunction, but not every wavefunction is an energy eigenfunction.

Nevertheless, it turns out that any such wavefunction ψ(x) can be written as a superposition of the eigenfunctions φ_n(x). That is, we can write

ψ(x) = ∑_na_nφ_n(x) ,

for some complex coefficients a_n, where

∑_n|a_n|² = 1 .

 

[masudr]

That's a good, thorough post, Eye_in_the_Sky. However, I would prefer to start my energy eigenvalues from E₁ and energy eigenfunctions from φ₁(x), because we are dealing with a particle in an infinite square well.

 

[Eye_in_the_Sky]

Yes, masudr, I see what you mean. Starting with n=1 (instead of n=0) makes the actual expressions for E_n and φ_n(x) somewhat easier on the eye, since in that case only an "n" (instead of an "n+1") will appear.