What's the physical meaning of an eigenvalue?

[seek]

Homework Statement

This isn't a homework problem, just something I've been trying to conceptualize for a while. Can anyone exemplify with a physical analog the concept of eigenstates? For example, I know that eigenvalues of variables with continuous spectra do not exist in the physical hilbert space, but I really can't get a working grasp of that concept. This would really help to solidify QM for me if someone can explain this, thanks.

 

[Ben Niehoff]

In some situations, you can have continuous spectra, such as the energy bands in crystals.

Eigenvalues are easiest to understand in terms of linear algebra. A square matrix represents a transformation on some vector space; the eigenvectors are the directions in which the matrix acts solely as a scaling transformation, and the eigenvalues are the corresponding scale factors.

I find it easiest to understand that first, and then look at Quantum Mechanics as a generalization of linear algebra to an infinite-dimensional Hilbert space. An operator in Hilbert space is in some sense an infinite-dimensional square matrix, and the eigenstates are infinite-dimensional column vectors.

Another way to think of it physically is in terms of resonance frequency and normal modes. Take a violin, for example. It has a fairly complicated shape. But for some frequencies of vibration, only one mode of oscillation is excited. This mode is an eigenstate; the frequency (or its square) is an eigenvalue.

The modes of vibration of some arbitrary shape can be mapped out by putting a thin layer of sand on the surface, and using a speaker-like device to drive vibrations at a particular frequency. At resonance frequencies, the sand forms a stationary pattern of gaps corresponding to the nodes of the standing wave pattern (or was it antinodes? I forget). The pattern you see is an eigenstate of that shape; the (square of the) frequency is its eigenvalue.

 

[turin]

You can consider a 3-D example. Consider a rotation about the z-axis (of some angle that is not a multiple of pi). e_z (or alternatively -e_z) is an eigenvector of this rotation with eigenvalue = 1. The reason is that the rotation does not change this vector.

If you restrict the vector space to a real field, then a rotation only has one eigenvector, and the eigenvalue is always = 1. However, if you allow a complex field, then there are two more orthogonal eigenvectors. Do you know what they are, and what are their eigenvalues? These other two eigenvectors and eigenvalues are actually important to particle physics.

 

[seek]

I just wanted to thank the both of you for your very helpful input. I've been a little busy, so I haven't had the time that I wanted to really go over your information. I'll be sure to do so though. Thank you :)

 

[amezcua]

seek Martin Schleske has done a lot of work on this connected with violins.Much easier to visualise when you see his moving violin plates.Quite frankly it all baffles me.Pretty pictures though.

 

[spal]

seek Martin Schleske has done a lot of work on this connected with violins.Much easier to visualise when you see his moving violin plates.Quite frankly it all baffles me.Pretty pictures though.

Martin Schleske