Difference between expectation value and eigenvalue

[Sara Kennedy]

There is another topic for this but I didn't quite see it and I don't know how I've gone so far through my course not asking this simple question. So what's the difference?

My thought process for hydrogen. I know it can have quantised values of energy, the energy values are the Eigen values of the wave state. There can be multiple Eigen values, E, with a corresponding wave states, Ψ.
Which can be of the form HΨ=EΨ.

If I want to find the expectation value, <H> this will give me the value of E that is most likely to be found every time I repeat a measurement on the system? The value I get for <H> will not change, but the value of E in the other equation can change.

Is my thought process correct?

 

[Nugatory]

The expectation value is the average that we would find if we were to make perform our measurement on a large number of systems, all prepared in the same way. That is not the same thing as the value we are most likely to find and it is not necessarily an eigenvalue; the average family has 2.5 children, but you will never ever find a family with 2.5 children - the number of children will always be some integer.

Suppose the quantum state is $\Psi=\alpha\psi_1+\beta\psi_2$. Here $\psi_1$ and $\psi_2$ are eigenfunctions of $H$ with the eigenvalues $E_1$ and $E_2$, meaning that they are solutions of the time-independent Schrödinger equation $H\psi=E\psi$: $H\psi_1=E_1\psi_1$ and $H\psi_2=E_2\psi_2$.

$\Psi$ is a solution of the time-dependent Schrödinger equation (which, strictly speaking, is the real Schrödinger equation) $H\psi=E\psi$ where $E$ is the energy operator $i\hbar\frac{\partial}{\partial{t}}$. If we measure the energy of this system, we will get one of two answers: $E_1$ with probability $|\alpha|^2$ and $E_2$ with probability $|\beta|^2$.

Now if we prepare a large number of systems in the state $\Psi$ and perform an energy measurement on each one, the average of all our measurements will come out to $E_1|\alpha|^2+E_2|\beta|^2$ - that's the expectation value.

 

[Sara Kennedy]

I read that a few times. Still clueless

 

[drvrm]

My thought process for hydrogen. I know it can have quantised values of energy, the energy values are the Eigen values of the wave state. There can be multiple Eigen values, E, with a corresponding wave states, Ψ.
Which can be of the form HΨ=EΨ.

You must think about the wave function of the physical system which are solutions of the Schrödinger equation -
H operating on wave function giving E times the wave function -
In general the wave function is/can be expanded in a complete orthonormal set of possible eigen functions and expectation value is the result of a measurement which can be any state of the system
having any of the possible eigen state -not a particular one - each particular value has finite probability of its occurrence.

 

[Sara Kennedy]

Suppose the quantum state is $\Psi=\alpha\psi_1+\beta\psi_2$. Here $\psi_1$ and $\psi_2$ are eigenfunctions of $H$ with the eigenvalues $E_1$ and $E_2$, meaning that they are solutions of the time-independent Schrödinger equation $H\psi=E\psi$: $H\psi_1=E_1\psi_1$ and $H\psi_2=E_2\psi_2$.

$\Psi$ is a solution of the time-dependent Schrödinger equation (which, strictly speaking, is the real Schrödinger equation) $H\psi=E\psi$ where $E$ is the energy operator $i\hbar\frac{\partial}{\partial{t}}$. If we measure the energy of this system, we will get one of two answers: $E_1$ with probability $|\alpha|^2$ and $E_2$ with probability $|\beta|^2$.

Now if we prepare a large number of systems in the state $\Psi$ and perform an energy measurement on each one, the average of all our measurements will come out to $E_1|\alpha|^2+E_2|\beta|^2$ - that's the expectation value.

Okay I see what you are saying now.

$E_1|\alpha|^2+E_2|\beta|^2$

So the expectation value is not actually going to be one of the values of E... say E1 or E2, but as you said... maybe 2.5 children. Its not the mode but the mean?

 

[Nugatory]

So the expectation value is not actually going to be one of the values of E... say E1 or E2, but as you said... maybe 2.5 children. Its not the mode but the mean?

Pretty much, yes.