Quantum mechanics: potential steps

[Martin89]

Homework Statement

The Attempt at a Solution

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Hi All,

I'm having trouble answering part (f) of the above question. I have managed parts (d) and (e) fine but am not sure how to proceed with part (f). I am pretty sure that the amplitude of the reflected wave in region 1 will be zero but I don't know how to show it.

Thanks in advance for any help!

 

[VKint]

Your solutions to (d) and (e) look correct, except for one small detail--I can't tell if the right-hands side of your last equation is $k_1 Te^{ik_1 L}$ or $k_2 Te^{ik_2 L}$. Obviously, one is correct and one is not. :)

If $Lk_2 = n \pi$, then $e^{iLk_2} = e^{-iLk_2} = \pm 1$. This should allow you to rewrite the boundary conditions at $x = L$ in a form that's easier to compare to the boundary conditions at $x = 0$---in fact, you should be able to eliminate $C$ and $D$ from the equations involving $A$ and $B$ altogether (although $T$ will still be present). If you can get this far, it should be pretty clear what to do next. (Hint: your intuition is correct. ;))

 

[Martin89]

Your solutions to (d) and (e) look correct, except for one small detail--I can't tell if the right-hands side of your last equation is $k_1 Te^{ik_1 L}$ or $k_2 Te^{ik_2 L}$. Obviously, one is correct and one is not. :)

If $Lk_2 = n \pi$, then $e^{iLk_2} = e^{-iLk_2} = \pm 1$. This should allow you to rewrite the boundary conditions at $x = L$ in a form that's easier to compare to the boundary conditions at $x = 0$---in fact, you should be able to eliminate $C$ and $D$ from the equations involving $A$ and $B$ altogether (although $T$ will still be present). If you can get this far, it should be pretty clear what to do next. (Hint: your intuition is correct. ;))

Thanks for the reply! I think I'm still missing something. Should k₁ = k₃ at x=0 or something?

 

[nrqed]

View attachment 226253

Thanks for the reply! I think I'm still missing something. Should k₁ = k₃ at x=0 or something?

You have not yet imposed that the wave functions must satisfy Schrodinger's equation. Impose that in regions I and III, what does that tell you about k1 and k3? (note that the energy of the particle is fixed and is assigned the value E in the question)

 

[Martin89]

You have not yet imposed that the wave functions must satisfy Schrodinger's equation. Impose that in regions I and II, what does that tell you about k1 and k3? (note that the energy of the particle is fixed and is assigned the value E in the question)

I've done as you said but I'm not sure what it shows me. K₁ and K₂ are different due to the potential in region 2. Looking at the problem physically, the Schrodinger equation must be identical in regions 1 and 3, and therefore K₁=K₃. Is this a correct assumption?

 

[nrqed]

I've done as you said but I'm not sure what it shows me. K₁ and K₂ are different due to the potential in region 2. Looking at the problem physically, the Schrodinger equation must be identical in regions 1 and 3, and therefore K₁=K₃. Is this a correct assumption?

Well, it would be better to show things explicitly. What condition do you get when you plug in the wave function of region I into Schrodinger's equation?

 

[Martin89]

Well, it would be better to show things explicitly. What condition do you get when you plug in the wave function of region I into Schrodinger's equation?

That is the condition for K₁. The expression for K₃ should be identical right?

 

[Dr Transport]

View attachment 226294

That is the condition for K₁. The expression for K₃ should be identical right?

do the calculation.......the answer will pop right out.

 

[Martin89]

do the calculation.......the answer will pop right out.

Ok so I've done the calculation and K₁ does indeed equal K₃. Thanks guys!!!