How Can the Triangular Potential Well TISE Be Simplified?

[QuantumJG]

Homework Statement

Show that the TISE expression you found in part (a):

I found it (a) to be:

$$\frac{d^{2} \psi}{dx^{2}} - \frac{2m}{\hbar ^{2}}(e \xi x - E) \psi (x) = 0$$

Show it (a) can be simplified to:

$$\frac{d^{2} \psi}{dw^{2}} - w \psi = 0$$

Homework Equations

w = z - z₀

where:

$$z = \left( \frac{2me \xi }{ \hbar ^{2} } \right)^{ \frac{1}{3} } x$$

$$z_{0} = \frac{2mE}{\hbar ^{2}} \left( \frac{ \hbar ^{2} }{ 2me \xi } \right)^{ \frac{2}{3} }$$

The Attempt at a Solution

So far I have found:

$$w = \left( \frac{2m}{ \hbar ^{2} e^{2} \xi ^{2} } \right)^{ \frac{1}{3} } (e \xi x - E)$$

But my problem is:

how do I convert:

$$\frac{d^{2} \psi}{dx^{2}}$$ to $$\frac{d^{2} \psi}{dw^{2}}$$, because I'm totally stuck on that.

 

[nickjer]

Well, you know w in terms of x. So what is dw in terms of dx? Then solve for dx^2, and plug it into your Schrödinger Eq.

 

[QuantumJG]

well,

$$dw = \left( \frac{2me \xi }{ \hbar ^{2} } \right)^{ \frac{1}{3} } dx$$

$$\therefore dw = z dx$$

But I'm confused as to how that's going to help.

 

[vela]

Homework Statement

Show that the TISE expression you found in part (a):

I found it (a) to be:

$$\frac{d^{2} \psi}{dx^{2}} - \frac{2m}{\hbar ^{2}}(e \xi x - E) \psi (x) = 0$$

Show it (a) can be simplified to:

$$\frac{d^{2} \psi}{dw^{2}} - w \psi = 0$$

Homework Equations

w = z - z₀

where:

$$z = \left( \frac{2me \xi }{ \hbar ^{2} } \right)^{ \frac{1}{3} }$$

$$z_{0} = \frac{2mE}{\hbar ^{2}} \left( \frac{ \hbar ^{2} }{ 2me \xi } \right)^{ \frac{2}{3} }$$

Are you sure this is right? Where did x go? And where did the cube roots come from?

The Attempt at a Solution

So far I have found:

$$w = \left( \frac{2m}{ \hbar ^{2} e^{2} \xi ^{2} } \right)^{ \frac{1}{3} } (e \xi x - E)$$

This seems unnecessarily complicated. Perhaps I'm missing something obvious, but if you compare the original and simplified equations, can't you see what w is equal to by inspection?

But my problem is:

how do I convert:

$$\frac{d^{2} \psi}{dx^{2}}$$ to $$\frac{d^{2} \psi}{dw^{2}}$$, because I'm totally stuck on that.

Use the chain rule:

$$\frac{df}{dx} = \frac{dw}{dx}\frac{df}{dw}$$

 

[nickjer]

well,

$$dw = \left( \frac{2me \xi }{ \hbar ^{2} } \right)^{ \frac{1}{3} } dx$$

$$\therefore dw = z dx$$

But I'm confused as to how that's going to help.

Knowing this and squaring it to get dx^2, you can replace the denominator in

$$\frac{d^2 \psi}{dx^2}$$

with that and get the 2nd derivative in terms of w. Then you will want to divide out the factor in front of your new 2nd derivative to get the 2nd equation you listed in your first post.

EDIT: The chain rule will also get you the same thing. But since w is linearly proportional to x, I just take the proportionality constant out and square it.

EDIT2: Also, the cube roots look right. They are needed to cancel out all the coefficients in the final form of the equation. If QuantumJG finishes this last step, then he should get a simple ODE in terms of w only and no other terms.

 

[QuantumJG]

Are you sure this is right? Where did x go? And where did the cube roots come from?

I just realized I forgot to put x into the equation for z

This seems unnecessarily complicated. Perhaps I'm missing something obvious, but if you compare the original and simplified equations, can't you see what w is equal to by inspection?

Use the chain rule:

$$\frac{df}{dx} = \frac{dw}{dx}\frac{df}{dw}$$

but then wouldn't:

$$\frac{d}{dx} \left( \frac{df}{dx} \right) = \frac{d}{dx} \left( \frac{dw}{dx}\frac{df}{dw} \right)$$

$$= \frac{df}{dw} \frac{d}{dx} \left( \frac{dw}{dx} \right) + \frac{dw}{dx} \frac{d}{dx} \left( \frac{df}{dw} \right)$$

 

[nickjer]

Use the chain rule on the last term:

$$\frac{dw}{dx}\frac{dw}{dx}\frac{d}{dw}\left(\frac{df}{dw}\right)$$

 

[QuantumJG]

Use the chain rule on the last term:

$$\frac{dw}{dx}\frac{dw}{dx}\frac{d}{dw}\left(\frac{df}{dw}\right)$$

Wait that's confusing, why did you put in that extra $$\frac{dw}{dx}$$?

 

[vela]

EDIT2: Also, the cube roots look right. They are needed to cancel out all the coefficients in the final form of the equation. If QuantumJG finishes this last step, then he should get a simple ODE in terms of w only and no other terms.

Oh, okay, I did miss something obvious.

 

[vela]

Wait that's confusing, why did you put in that extra $$\frac{dw}{dx}$$?

That's the one from the first differentiation. Because dw/dx is a constant, you get

$$\left(\frac{d}{dx}\right)^2 = \frac{dw}{dx} \frac{d}{dw}\left(\frac{dw}{dx} \frac{d}{dw}\right) = \left(\frac{dw}{dx}\right)^2 \frac{d^2}{dw^2}$$