Why do expansion coefficients give probability of energy in stationary states?

[RPI_Quantum]

In dealing with solutions to the Schroedinger equation, I have come across the so-called expansion coefficients (c n ). I have read that the square of the coefficient is the probability density of an allowed value of E.

How is there a probability of any given value of E? I know that in a stationary state, there are definite values of E, so how is there any probability involved?

 

[quantumdude]

In dealing with solutions to the Schroedinger equation, I have come across the so-called expansion coefficients (c n ). I have read that the square of the coefficient is the probability density of an allowed value of E.

Not the probability density. |c_n|² is the probability that the particle will be found in state n.

How is there a probability of any given value of E? I know that in a stationary state, there are definite values of E, so how is there any probability involved?

States need not be stationary. You can have a superposition of states with different energies. Of course, if the state is of a single energy, then the probability that the system will be found with that one energy is 1.

BTW, am I to take it from your name that you're taking QM at Rensselaer Polytechnic Institute? If so, who's teaching it these days?

 

[jtbell]

You can combine stationary states (in a linear combination) to get a non-stationary state in which the probability distribution "moves around". That is, the expectation value of x, <x>, is not constant, but varies with time.

A simple example is to take two stationary states of the one-dimensional "particle in a box" (infinite square well), say the two with lowest energy, and add them together:

$$\Psi (x, t) = \frac {1}{\sqrt {2}} ( \Psi_1 (x, t) + \Psi_2 (x, t))$$

(the $$\sqrt {2}$$ is to make the sum normalized, provided the two original wave functions are normalized to begin with)

Calculate the probability distribution for this wave function and you'll see that it oscillates with frequency $$(E_2 - E_1) / h$$.

Hmm, Tom can type faster than I can, apparently!

 

[dextercioby]

Don't tell me it took u 4hrs to type 15 lines...

Daniel.

 

[jtbell]

Don't tell me it took u 4hrs to type 15 lines...

Hey, I sweated blood over that LaTeX!

(besides, at 1:30AM I can't read very well... :zzz: )

 

[RPI_Quantum]

Thanks guys. And Tom, Professor Tim Hayes is the QM professor. He has been teaching it since before I got here (I am a sophomore). I take it you are an alumnus?

 

[quantumdude]

Thanks guys. And Tom, Professor Tim Hayes is the QM professor. He has been teaching it since before I got here (I am a sophomore). I take it you are an alumnus?

Yep, I took it with Gwo-Ching Wang.