Expectation value of energy in TISE

[yosimba2000]

If Eψ = Hψ, then why is expected energy ∫ψ*Hψ dx? It makes more sense if I see the ψ on the right side of H as the ψ in ∫Q(ψ*ψ) dx, where Q is some quantity we want to measure the expectation of.

But if true, then since H is defined as (h²/2m) (d²/dx) + V, then what does it mean to calculate the expectation value of H? H has no argument for the second derivative term.

Or another way is, since Eψ = Hψ, then E = H, then H = (h²/2m) (d²/dx) + V = E. How can you get a number from H if you don't take the derivative of anything?

 

[Nugatory]

Or another way is, since Eψ = Hψ, then E = H

That doesn't follow at all. For a trivial counterexample, consider the eigenvalue equation $A\psi=\alpha\psi$ where $\alpha$ is a constant and $A$ is the matrix
$$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$
One solution of this equation (there's another one, and you might find it worth your time to try to find it) is $\alpha=1$ and $\psi=\begin{bmatrix} 1 \\ 1 \end{bmatrix}$
but you certainly wouldn't conclude from that that $A=\alpha$. The Schrodinger equation is similar - on one side you have an operator acting on a function, and on the other side you have a constant multiplying that function, and solving it is a matter of finding a function and a constant that makes the two sides come out equal.

If Eψ = Hψ, then why is expected energy ∫ψ*Hψ dx? It makes more sense if I see the ψ on the right side of H as the ψ in ∫Q(ψ*ψ) dx, where Q is some quantity we want to measure the expectation of.

$E$ is just a constant so we can take it out of the integral. We have:
$$\int\psi^*H\psi dx = \int\psi^*E\psi dx = E\int\psi^*\psi dx$$
where I used the fact that $E$ and $\psi$ are solutions of the Schrodinger equation in the first step. Now, what is the value of $\int\psi^*\psi dx$?

If this does not all make sense to you now, you may need a bit more math before you take on quantum mechanics. You'll need some linear algebra and differential equations beyond what's usually taught in first-semester intro classes.

 

[yosimba2000]

That doesn't follow at all. For a trivial counterexample, consider the eigenvalue equation $A\psi=\alpha\psi$ where $\alpha$ is a constant and $A$ is the matrix
$$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$
One solution of this equation (there's another one, and you might find it worth your time to try to find it) is $\alpha=1$ and $\psi=\begin{bmatrix} 1 \\ 1 \end{bmatrix}$
but you certainly wouldn't conclude from that that $A=\alpha$. The Schrodinger equation is similar - on one side you have an operator acting on a function, and on the other side you have a constant multiplying that function, and solving it is a matter of finding a function and a constant that makes the two sides come out equal.$E$ is just a constant so we can take it out of the integral. We have:
$$\int\psi^*H\psi dx = \int\psi^*E\psi dx = E\int\psi^*\psi dx$$
where I used the fact that $E$ and $\psi$ are solutions of the Schrodinger equation in the first step. Now, what is the value of $\int\psi^*\psi dx$?

If this does not all make sense to you now, you may need a bit more math before you take on quantum mechanics. You'll need some linear algebra and differential equations beyond what's usually taught in first-semester intro classes.

I'm not far in linear algebra but I've taken Calc1-3 and Diff Eq, so I don't quite get eigenvalues, or that example. $\int\psi^*\psi dx$ should be 1, or it would be if it was normalized. How do we know if it is normalized right now?

But I'm still not understanding why expectation of energy is ∫ψ*H(ψ) dx. So H is supposed to be a function, and its input is ψ, right? So if Eψ = H(ψ), then energy should be E = H(ψ)/ψ, and expected energy would be <E> = ∫(ψ*ψ)(H(ψ)/ψ) dx?

 

[Nugatory]

So H is supposed to be a function, and its input is ψ, right?

H is an operator, something whose input is one function and whose output is another function. When we're solving the time-independent Schrodinger equation, we looking for a function $\psi(x)$ and a number $E$ such that if we provide the function $\psi(x)$ as input to the operator $H$, the operator will output the function $E\psi(x)$.

The first step in evaluating the integral in your first post is to replace $H\psi$ with whatever function is the output of $H$ when the input is $\psi(x)$. That leaves you an integrand that is just a function of $x$ so you can do the integration over $dx$.

But I'm still not understanding why expectation of energy is ∫ψ*H(ψ) dx.

The expectation value of any observable, including energy, is given by $\int \psi^*\hat{O}\psi$ where $\hat{O}$ is the Hermitian operator corresponding to that observable. The Hamiltonian is that operator for energy.