A Step potential question

In summary, the conversation discusses a question on a past exam involving a particle incident on a step potential at x=0 with total energy E less than the height of the step U. The particle has wavefunctions with equal k values on both sides of the step. The conversation then explores how k must be related to E and U and determines the ratio E/U. It also discusses how the wavefunction can be put in a given form and concludes with a discussion on the possibility of E=U/2 in quantum mechanics.
  • #1
heardie
24
0
Hi...I've come across this question on a past exam, and I can't seem to resolve it!

A particle is incident on a step potential at x=0. The total energy of the particle E, is less then the height of the step, U. The particle has wavefunctions
[tex]$\begin{array}{l}
\psi (x) = \frac{1}{2}\{ (1 + i)e^{ikx} + (1 - i)e^{ - ikx} \} ,x \ge 0 \\
\psi (x) = e^{ - kx} ,x < 0 \\
\end{array}$
[/tex]

Note that k is the same on both sides of the step
a) ….
b.) How must k be related to E and U on both sides of the step and determine the ration E/U

Well on the left, it is just a free wave-function where k=[tex]$k = \frac{{\sqrt {2mE} }}{\hbar }$
[/tex]
On the right k= [tex]$k = i\kappa = \frac{{i\sqrt {2m(U - E)} }}{\hbar }$
[/tex]
Since the k’s are equal this immediately implies that U=0, does it not? In which case the ratio E/U is infinite…
 
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  • #2
heardie said:
Since the k’s are equal this immediately implies that U=0, does it not? In which case the ratio E/U is infinite…
No, but you are thinking along the right lines. You have recognized that the requirement on k puts strong restriction on ... something. Attack the problem firstly as if k were different in the two regions, and then see what conditions have to be met so that the wavefunction can be put in the given form. (Biggest hint: Notice that there is an i in the exponent in one region and not in the other.)

One thing I just noticed: it appears to me that the particle is incident from the right so you may have your suggestions backwards.
 
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  • #3
Ok so if I call the region II solution k'
[tex]$k' = \frac{{\sqrt {2m(E - U)} }}{\hbar }$[/tex]

Since U > E this is complex.
[tex]$k' = \frac{{i\sqrt {2m(U - E)} }}{\hbar } = i\kappa $
[/tex]

This leads to the exponential solution
[tex]$e^{ - \kappa x} $
[/tex]

If I then impose
[tex]$\kappa = k$ [/tex] i get
[tex]$\begin{array}{l}
\frac{{\sqrt {2mE} }}{\hbar } = \frac{{\sqrt {2m(U - E)} }}{\hbar } \\
\sqrt E = \sqrt {U - E} \\
E = U - E \\
2E = U \\
E = \frac{U}{2} \\
\end{array}$
[/tex]

oh hey...i definatly did that yesterday...musta been dropping a negative somewhere!

Ok, seems my latex sucks but how is E=U/2??

Cheers!
 
  • #4
heardie said:
... how is E=U/2??
That was my initial gut feeling about it. Are you asking how it is possible, or are you asking if I think this is correct? I think it is probably correct (without actually combing through your calculations).
 
  • #5
Turin - I don't think I would ever ask if something is possible in QM, per se, since things that seem impossible always seem to fall out of QM! So much more to learn as well - makes me wonder what other things (classically impossible) will occur
 

1. What is a step potential?

A step potential is a sudden change or discontinuity in a physical or electrical quantity, such as potential energy or voltage. It is often represented by a sudden jump or step in a line or graph.

2. What is the significance of a step potential in science?

A step potential is important in understanding the behavior of particles or waves when they encounter a sudden change in their environment. This can have implications in fields such as quantum mechanics, electromagnetics, and fluid dynamics.

3. How is a step potential calculated?

The calculation of a step potential depends on the specific variables involved, such as the magnitude of the change and the initial and final values. In general, it involves using mathematical equations or formulas to determine the change in potential energy or voltage.

4. What are some real-world examples of a step potential?

One example of a step potential is the sudden change in voltage experienced by a particle in an electrical circuit when it passes through a resistor. Another example is the sudden change in potential energy experienced by a particle as it moves from one medium to another, such as from air to water.

5. How does a step potential affect particle behavior?

A step potential can cause particles to behave differently than they would in a uniform environment. Depending on the magnitude of the change, it can cause particles to reflect, transmit, or be absorbed at the step. It can also affect the energy levels and trajectories of particles, leading to important consequences in various scientific phenomena.

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