Understanding Particle Superposition in the Schrodinger Equation

[karkas]

Hello, 've been progressing through my self-studying of the Schrödinger Equation in both its time-dependent and independent forms, and I have come across an unknown term.

Super Position ( in my book it's translated in greek literally superposition = υπέρθεση)

My guess so far is that a superposition is when a particle is described by two wavefunctions, which happen to be two eigenfunctions $\psi_n$with the same (perhaps with different, {not sure there} ) eigenvalues En. Am I correct? If not, please enlighten me :)

 

[malawi_glenn]

well it does not have to be a particle either! :-)

But consider deuterium, a bound proton - neutron state. It's state function is a linear combination of two terms: http://en.wikipedia.org/wiki/Deuterium#Approximated_wavefunction_of_the_deuteron

 

[karkas]

I think "Linear Combination of Wavefunctions" was the term I was looking for, eh?

 

[confinement]

I think "Linear Combination of Wavefunctions" was the term I was looking for, eh?

The term you are looking for is "linear combination of Energy eigenstates." For example, take the case of a particle in a 1D box of width L. The energy eigenfunctions are:

$$\phi_n = A sin(\frac{n \pi x}{L})$$

Where A is a normalization factor. Just because a particle is in this box, does not mean that it is one of the states, those are only the states with definite well defined energy. A particle could be in a super position of energy eigenstates:

$$\psi= B sin(\frac{\pi x}{L}) + C sin(\frac{2 \pi x}{L})$$

where a condition on B and C is to normalize the wavefunction, as usual. Notice that the two states which are involved are the n = 1 state (the B term) and the n = 2 state (the C term). Now when we measure the energy of a particle in this state we do not know whether
you will get n = 1 or n = 2 but we can calculuate the probabiity of either!

 

[karkas]

Thank you! Really nice explanation there!