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Topic: Drawing Wave Functions from Potential Wells  (Read 16616 times)

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Drawing Wave Functions from Potential Wells
« on: March 25, 2010, 03:37:48 PM »
Has anyone got any tips or any links on how to draw the corresponding wave funtions for potential wells at different energies?

Really having problems getting it even though it's probably simple enough to those who understand it, for example stuff like this;

Am I right in thinking that when the kinetic energy rises the wave will be decreasing i.e. the potential energy?

Any help is very much appreciated

tamim83

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Re: Drawing Wave Functions from Potential Wells
« Reply #1 on: March 26, 2010, 09:14:55 AM »
You can divide the potential well into two regions.  Region 1 is E<b and Region 2 is E>b.  For region 1, think about tunneling.  The wavefunction will look like a particle in a box wavefunction until it hits the barrier.  Then it exponentially decays.  You should get some wave amplitude on the other side but the amount will depend on the energy.  Higher energy means more transmission through the barrier.

For region 2, there is a possibility that the wave gets reflected, this probability decreases as E >> b.

Check out http://en.wikipedia.org/wiki/Rectangular_potential_barrier

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Re: Drawing Wave Functions from Potential Wells
« Reply #2 on: March 26, 2010, 12:35:20 PM »
You can divide the potential well into two regions.  Region 1 is E<b and Region 2 is E>b.  For region 1, think about tunneling.  The wavefunction will look like a particle in a box wavefunction until it hits the barrier.  Then it exponentially decays.  You should get some wave amplitude on the other side but the amount will depend on the energy.  Higher energy means more transmission through the barrier.

For region 2, there is a possibility that the wave gets reflected, this probability decreases as E >> b.

Check out http://en.wikipedia.org/wiki/Rectangular_potential_barrier

Ok I've done a quick sketch of what I think the waves should look like then and am I getting it right or am I still way off?

For E1 I've done it so that it doesn't have enough energy to tunnel to the other side and for E2 I've drew it so that it does

What should E3 look like?

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Re: Drawing Wave Functions from Potential Wells
« Reply #3 on: March 28, 2010, 02:12:29 PM »
Coming back to this....

Should E1 look rather more like something like this and should E3, since it has a higher energy than the potential, just go over it (starting at 0,∞ and ending at a,∞  of course, only noticed I didn't draw it like that just now) or should it have two nodes (i.e. cross the x axis)??

Again, thanks for any help

Grundalizer

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Re: Drawing Wave Functions from Potential Wells
« Reply #4 on: March 28, 2010, 06:31:22 PM »

Basically keep adding 1/2 wavelength in each time.

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Re: Drawing Wave Functions from Potential Wells
« Reply #5 on: March 28, 2010, 07:47:10 PM »
Yeah I can understand that for when there's no barrier but once a potential barrier is introduced into the mix it sends me all haywire and I don't know what shape of a function to draw

FreeTheBee

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Re: Drawing Wave Functions from Potential Wells
« Reply #6 on: March 29, 2010, 02:35:34 AM »
With the extra barrier the node thing still holds, since the energy of the particle is related to the curvature of the wave function.
I think for the lowest energy, you could see each side as a box with infinite height, since the particle has very low energy and its wave function will therefore not extent far into the middle part. So the particle will be in one box only.
At an intermediate energy the wave function (with shorter wavelength now) starts to extend into the high energy part, since the potential energy there is not infinite. The function will decay exponentially in that mid section. I would choose not to have the function extent far enough yet to feel a similar function on the other side.
At very high energy the function (even shorter wave length now) extends so far into the mid section that it reaches the other low energy side or feels a similar wave function from the other side and you can have tunneling. I think you would then get a sine like wave on both sides of the barrier which do not hit this finite barrier at 0, but higher. In the middle part I think it would look like the sum of the exponentially decaying wave functions from either side.

It's been a long time since dealing with particles in boxes, so I am not entirely sure.

http://en.wikipedia.org/wiki/Finite_potential_well

FreeTheBee

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Re: Drawing Wave Functions from Potential Wells
« Reply #7 on: March 29, 2010, 05:19:57 AM »
Some extra remarks I didn't think of earlier. Exponential function go to 0 very slowly of course, so even at low energies the wave function could extend far enough to feel each other. Then, at low energy tunneling will be possible, but with low probability.
Also, even at the lowest energy I guess there should be a wave function in both boxes, except only one of them will be occupied.

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Re: Drawing Wave Functions from Potential Wells
« Reply #8 on: March 29, 2010, 04:16:09 PM »
Ok thanks for all that Freethebee. Here's another attempt.

I've drawn E1 where it doesn't have the energy to tunnel, E2 where it does have the energy to tunnel and E3 where it is pretty much unaffected by the potential increase

For example, for E2 where it does not have the energy to tunnel, does it just look like what I have drawn for E1 but with a bit longer exponential decay to zero or does it have to cross the x axis and keep on decaying and end up having negative energy when it hits the infinite potential wall at a like I've drawn below?

Am I getting any closer to the right answers here or am I making any progress at all?

Here's another example of a well

Should the functions look anything like these attempts?

For 2 and 3 I've drawn the amplitude lower once it tunnels a bit into the barrier because the probability of finding it there gets lower but I'm not sure for 2 especially if functions are even allowed to decay in that direction?? i.e. upwards to zero

Thanks to anyone who is taking the time to look at my attempts.

FreeTheBee

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Re: Drawing Wave Functions from Potential Wells
« Reply #9 on: March 30, 2010, 03:03:23 PM »
I was searching a little bit, to make sure I wasn't talking nonsense here and found this pdf. Unfortunately the source is bit unclear but it seems to make sense,
http://s3.amazonaws.com/cramster-resource/8608_n_21711.pdf
On page 13 is a sketch of a similar system.

On the 2nd sketch of E2 you mention the energy going down. Remember that the value of the wave function itself does not give you the energy. The square gives you the probability of finding the particle in that position and the curvature is related to the kinetic energy. The exponential decay in the barrier basically means the probability of finding the particle gets closer to 0 the deeper into the barrier you go, it shouldn't become negative. If it would go from positive to negative, the square would become bigger again, meaning the chance of finding the particle deep in the barrier increased.

I am not sure how the wave would look with the sloped potential barrier. I guess it decreases quicker, but how I wouldn't know. The risk of me writing rubbish is starting to increase here (exponentially), so I'd better not make any suggestions

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Re: Drawing Wave Functions from Potential Wells
« Reply #10 on: March 30, 2010, 05:35:53 PM »
I was searching a little bit, to make sure I wasn't talking nonsense here and found this pdf. Unfortunately the source is bit unclear but it seems to make sense,
http://s3.amazonaws.com/cramster-resource/8608_n_21711.pdf
On page 13 is a sketch of a similar system.

On the 2nd sketch of E2 you mention the energy going down. Remember that the value of the wave function itself does not give you the energy. The square gives you the probability of finding the particle in that position and the curvature is related to the kinetic energy. The exponential decay in the barrier basically means the probability of finding the particle gets closer to 0 the deeper into the barrier you go, it shouldn't become negative. If it would go from positive to negative, the square would become bigger again, meaning the chance of finding the particle deep in the barrier increased.

I am not sure how the wave would look with the sloped potential barrier. I guess it decreases quicker, but how I wouldn't know. The risk of me writing rubbish is starting to increase here (exponentially), so I'd better not make any suggestions

Ok thanks for all your help