Which ψ do I use for the Expectation Value ?

[SpaceNerdz]

I have to calculate the Expectation Value of an Energy Eigenstate : < En >
The integral is ∫ ψ* En ψ dx

I have :
A ) ψ = √L/2 sin nπx/L , a single standing wave of the wave function
B ) ψ = BsinBcosD , the wave function of the particle
C ) ψ = ΣCn ψn = C , sum of all the standing waves

Which ψ do I use ? Why ? What's wrong with using the other two ?

 

[PeterDonis]

Which ψ do I use ?

The short answer is, whichever one corresponds to the energy eigenstate you are trying to calculate the expectation value of. But first you need to explain where you are getting these three functions from. What is the physical situation you are asking about? An electron in an atom? A free particle? What?

 

[Demystifier]

In other words, wave function is contextual.

 

[kith]

I have to calculate the Expectation Value of an Energy Eigenstate : < En >

That's not the correct terminology. There is no expectation value "of a state" but only an expectation value of a physical quantity (like energy, position, momentum, spin).

If you calculate the expectation value of energy, you know what to expect in the lab if you repeat the same measurement many times and average over the outcomes. What do I mean with "the same" in this sentence? You need to measure the same physical quantity (energy in this case) and you need to make sure that the system is in the same state $\psi$ for all measurements. So of course, the expectation value of a physical quantity depends on which state $\psi$ your system is in.

The expectation value of energy is often denoted as $\langle H \rangle_{\psi}$ where $H$ stands for the energy operator. Your notation $\langle E_n \rangle$ doesn't make sense, because it specifies only the state (which is $E_n$) but not the physical quantity of which you want to calculate the expectation value. You may say that it is obvious that you want to calculate the expectation value of energy but we could as well calculate the expectation value of position for your state $E_n$, which would read like this: $\langle X \rangle_{E_n}$.

The integral is ∫ ψ* En ψ dx

This formula is only correct if ψ is an energy eigenstate. Where does the formula come from? Has it been derived? Please give context and state your knowledge level.