00:01
In this problem, we're given that the wave function is a superposition of energy eigenstates, in this case, phi 2 and 54, and then equals 2 and equals 4, of a one -dimensional box of length a.
00:15
Now, the first part is to show that even though 52 and 54 energy eigenstates, psi -x is not.
00:26
So we are given that this is true.
00:29
Now drop the x, just for simplicity, say, in writing.
00:33
So we have this, and now let's act by the hamiltonian on the wave function.
00:46
So e2, psi 2, minus square of three quarters, e4, phi, four.
01:04
Now, for any box, whatever it's nature, the energy states, energy eigenvalues are not the same.
01:15
So this is not equal to some e -si.
01:21
Can't factor out.
01:22
You two and e4 are different, so how can you factor it out? you can't.
01:27
So that tells me if it's not this, then that means the wave function is not an energy eigenstate itself.
01:38
Part b wants to show that it is normalized.
01:42
Making use to the orth abnormal nature of the energy eigenstates, all the fies.
01:54
So 0 to a, phi star x, psi n x, so m and n d x is equal to chronicer delta, mn.
02:11
This is equal to 1 when m is equal to n, and 0 when m is not equal to n.
02:15
It's chronicer delta.
02:18
Okay, so we're given that fact that they are orthogonal, but they're also, they're also are normalized.
02:29
Okay, so let's do, that's calculate, let's look at the expression that should turn out to be one for the wave function si, psi star, si, tx, zero to a, one square root, one over four, sci two star, minus square root three over four, five, four star, one over square root at one over four, five, five, two, square root 3 over 4 54 dx that's putting in the actual expressions that we're given now it's just a matter expanding everything out so just like you'd have a plus b times c plus d you got to do all the terms so first one we get one quarter zero to a five two star 52 d x and we're just going to do the first term times each of the terms on the right then we'll do the second term minus square root of 3 over 4, 0 to a, psi 2 star, 54, dx, minus square root of 3 over 4.
04:07
Now we're doing the second term, 0 to a, 54 star, side 2, dx, and now we have the last term, 3 quarters, 0 to a, 5 4 star, 5, 4, x so that's expanding it all out but we have the end it doesn't matter this is true this is just a one or a zero so i could take the complex conjugate it i get the same results so phi four star phi two is the same as phi two star five four so they're both zero okay so the first integral here is one second integral, zero.
05:09
Third integral, zero.
05:11
That's the orthogonality is equal to one.
05:14
So we get one quarter plus three quarters, which i think is one.
05:21
So as required.
05:25
Or as requested.
05:28
That's what they wanted us to prove, show that it was.
05:31
So there we have it.
05:32
So it is what was generated, what was given as true position was normalized.
05:39
Part c, now you have a large collection of the same system, and you're going to take measurements on that large collection.
05:50
And they ask us what energy measurements could we get, and how many times we expect to see that particular value.
06:02
Okay.
06:04
So when we get the probability, that's what telling us.
06:09
If we get probability of one quarter, as we're going to get in the first case, that's 25 % of the time.
06:17
So if we had 1 ,000 measurements, we'd expect 250 of those, 25 % to be of that particular energy eigenvalue.
06:28
So that's how we use the probability when we have many, many measurements of that, not of the single system, but of that copies, 1 ,000 copies of that system.
06:42
Then you're taking a thousand measurements.
06:44
If you took the measurement on the same system, once it collapses into one state, into an eigenstate, it's done.
06:50
You take all the measurements you want.
06:51
You're going to get the exact same value.
06:54
Two measurements, two billion measurements.
06:56
It's going to be the same.
06:57
That's the way it goes.
06:58
So it's got to be on a thousand copies, not on one system a thousand times.
07:04
Okay, so let's get the probability of being, measuring energy e2.
07:15
That is the modular square, of the integral, because in general, these, remember, these values, when we do these, could be complex.
07:28
That's why we use the, that's why we use, we're not given any specification.
07:33
That's why we're always, even if it's a real function, if you're doing, even if you're doing something where they give you a real function, you still want to write this way.
07:40
Because this reminds you because also those normalization constants, yeah, so they've chosen one quarter.
07:47
Does it have to be? could there be complex aspect of this that when you take the modulus squared of it goes away, surely...
