Why do SHM solutions have no imaginary values?

[AdrianMay]

Hi all,

The textbook simple harmonic oscillator solution has no imaginary values. True or false? If true, why not? Most of the time you get something like XXX.exp[i(Et-p.x)].

I thought maybe it was a superposition of states such that the complex parts cancel, but in that case they'd both have the same energy and you'd probably mangle them into sum and difference form so as to get different eigenvalues.

Confused,
Adrian.

 

[weejee]

Due to the factor exp(-iEt/hbar) associated with time evolution, the full time-dependent wavefunction is always complex.

However, for a given time, an eigenstate can be written as a real function of position up to a complex phase factor. The reason for this has to do with time-reversal invariance.

 

[AdrianMay]

OK that's helpful, but nevertheless, for a square well, even a snapshot at a given instant in time has exp(ip.x) in there (doesn't it?) so there's no moment at which it all has the same phase. That seems to be different for this harmonic oscillator and I don't see why.

Adrian.

 

[weejee]

OK that's helpful, but nevertheless, for a square well, even a snapshot at a given instant in time has exp(ip.x) in there (doesn't it?) so there's no moment at which it all has the same phase. That seems to be different for this harmonic oscillator and I don't see why.

Adrian.

For a square well, it is sines and cosines. If we talk about free particles, however, it has the problem you've just pointed out. Still, since exp(ipx) and exp(-ipx) have the same energy, we can construct purely real eigenstates, which are cos(px) and sin(px).

 

[AdrianMay]

So would I be right in saying that if the phase depends on position then it's a traveling wave, but for standing waves (SHO, square well or whatever) the phase only depends on time so you can factor it out leaving all the interesting stuff behind as real?

Adrian.

 

[weejee]

So would I be right in saying that if the phase depends on position then it's a traveling wave, but for standing waves (SHO, square well or whatever) the phase only depends on time so you can factor it out leaving all the interesting stuff behind as real?

Adrian.

I think you are right. Still, whenever there is a time-reversal invariance, waves that travel forward and backward (with momentums p and -p) should have the same energy, so that we can make a purely real energy eigenfunctions by making suitable linear combinations of them.

Therefore, the most general condition for being able to construct a complete set containing only real eigenfunctions, is the time-reversal invariance, when there is no spin dependence in the Hamiltonian. If there is spin-orbit coupling, it is not necessarily true.