Find normalised linear combinations that are orthogonal

[g.lemaitre]

Homework Statement

I'm a little weary of posting this in this forum. If I post it in the math section it will be answered in about 30 min whereas here it might take about 5 hours, but we'll see.

Homework Equations

The Attempt at a Solution

Number one, I'm not exactly sure how they get from
$$\int \phi_1^* (c_1\phi_1 + c_2\phi_2)d\tau = c_1 + c_2 d$$

I think it's because

$$\phi^*\phi = 1$$ but I'm not sure.
Number two, I don't understand the following step:
$$\int (c_1\phi_1 + c_2 phi_2)*(c_1 \phi_1 + c_2\phi_2)d\tau = c_1^2 + c_2^2 + 2dc_1c_2$$

why does $\phi$ disappear?
I figure that it must have something to do with the fact that $\phi_1$ is orthogonal which means it = 0
Number three, I can't get step 3. I put the equations as follows:
$$c_1 + c_2d = 0<br /> c_1^2 + c_2^2 + 2 dc_1c_2 = 1<br /> c_1 = -c_2d$$
therefore
$$(-c_2d)^2 + c_2^2 + 2d(-c_2d)c_2 = 1<br /> (-c_2d)^2 + c_2^2 - 2(c_2d)^2 = 1$$
And then I can go no further.
Number four, I don't understand how
$$\int (\phi_1 + \phi_2)*(c_1\phi_1 + c_2\phi_2)d\tau$$

simplifies to
$$(c_1 + c_2)(1+d)$$
Number five, what do they mean by 2 and 5 gives
$$\frac{(\phi_1 - \phi_2)}{\sqrt{2-2d)}}$$
As you can see I'm real clueless with regards to this stuff. I've got a private tutor lined up but I won't be able to meet with him until sometime next week.

 

[Aimless]

From the first sentence of the problem: "$\phi_1$ and $\phi_2$ are normalized eigenfunctions..."

What is the definition of normalized? Specifically, how does the fact that $\phi_1$ and $\phi_2$ are normalized relate to the integrals $\int \phi_1^*\phi_1 d\tau$ and $\int \phi_2^*\phi_2 d\tau$?

Answer that question and that should tell you everything you need to know to work through the derivation.

 

[g.lemaitre]

Normalized means = 1. I already knew that. Still clueless.

 

[Aimless]

What, specifically, is equal to 1?

Edit: Hint - Your statement in the original post that $\phi_1^* \phi_1 = 1$ is not quite correct. What is the correct equation?

 

[g.lemaitre]

$$\phi_1 \phi_2$$

 

[Aimless]

Not true. Nothing in the problem set up indicates $\phi_1 \phi_2 =1$ in general. The correct relation between the two is $\int \phi_1^* \phi_2 d\tau = d$.

So, I think I'm seeing several issues here that are causing you problems. First, you need to be a lot more careful about the difference between $\phi$ and $\phi^*$. The two are not the same, and which terms have the * and which terms don't matters.

Second, you need to go back and review the definition of normalization. I think you have assumed that it means something that it doesn't mean, and that's why you can't evaluate the integrals above.

Third, slow down, take a bit more time, and don't skip any steps when you are working through the problem. Also, when replying on this thread, be as specific as you can; it'll make it easier to diagnose where your misunderstanding is coming from.

To work through each of the integrals in the derivation above, you need to know three different quantities:

$\int \phi_1^* \phi_1 d\tau = ?$
$\int \phi_2^* \phi_2 d\tau = ?$
$\int \phi_1^* \phi_2 d\tau = ?$

Each of the integrals in the derivation can be broken up into pieces containing those three terms.

 

[g.lemaitre]

I think I'll just wait until I meet with that private tutor. My problems are just so overwhelming. I think maybe I zipped through the calc, linear algebra and DE prerequisites too quickly. Maybe I'm trying QM without sufficient knowledge.

 

[Aimless]

Actually, the math in this particular derivation is quite simple, once you have all of your definitions straight. That's why I was asking you about what normalization meant. I don't think this particular derivation is beyond you, I just think you need to slow down, take your time with it, and review the basics.

As another hint: for a normalized (1d) wave function in the position representation, $\phi(x)$, the probability of finding the particle between the points $x_1$ and $x_2$ is given by $P[x_1<x<x_2]=\int_{x_1}^{x_2} \phi^*(x) \phi(x) dx$.

If I take $x_1$ to $-\infty$ and $x_2$ to $\infty$, what's the probability of finding the particle between $x_1$ and $x_2$? That is,

$P[-\infty<x<\infty]=\int_{-\infty}^{\infty} \phi^*(x) \phi(x) dx = ?$

Don't try to solve the integral, just use common sense. What's the probability of finding the particle if we search the entirety of the real line?

 

[g.lemaitre]

as for the odds of finding a particle between negative infinity and plus infinity it has to be 1, nothing can exist beyond the edge of infinity since there is no edge to infinity. I certainly hope the math is not beyond me.

 

[Aimless]

Exactly. $P[-\infty<x<\infty]=\int_{-\infty}^{\infty} \phi^*(x) \phi(x) dx = 1$. That is the definition of what it means for a wave function to be normalized.

So, what does that statement imply for the three integrals needed for this problem?

$\int \phi_1^* \phi_1 d\tau = ?$
$\int \phi_2^* \phi_2 d\tau = ?$
$\int \phi_1^* \phi_2 d\tau = ?$

 

[g.lemaitre]

I'm guessing they equal 1.

 

[Aimless]

Not all of them. :) One of the three is given to you in the problem. The other two equal 1.

 

[g.lemaitre]

$\int \phi_1^* \phi_2d\tau = 0$ because it's orthogonal, right?

 

[Aimless]

Nope. Actually, that's the point of the problem. $\phi_1$ and $\phi_2$ are not orthogonal, but rather integrate to a constant, $\int \phi_1^* \phi_2 d\tau = d$. The question is, given that the are not orthogonal, how would you go about constructing orthogonal functions from them.

That is, you are trying to find some $\bar{\phi}_2 \equiv c_1 \phi_1 + c_2 \phi_2$ such that $\int \phi_1^* \bar{\phi}_2 d\tau = 0$.

 

[g.lemaitre]

I have to leave now and I will return to this problem after about 20 hours have passed. I appreciate you helping me.