Particle in a box question in classical mechaincs

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Taylor_1989
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Hi guys, I am having an issue with what my lecture is saying in these slides, I have attched my slides below.
Lecture slide diagram of a classical two-state potential V(x): a central barrier of height Δ between two walls that rise to infinity, with a marked energy level E and the x axis.

Handwritten diagram of a particle in a box with labels L, barrier and R, showing green horizontal energy levels across three regions, with the caption "any energy E>0 is valid".


Two phase-space diagrams of a particle in a double box with a barrier: left for E < Δ with closed loops confined to the L and R boxes, right for E > Δ with a larger loop spanning both boxes through the barrier region; axes x and p, with a bracket labelled √(2mE).


Here is my issue. I am very confused by the ##E<\Delta ## Beacuse I can't see how this has any momentum as it would produce an imagery number. And ye say that ##E<\Delta ## cannot be a solution yet on these diagrams he show the particle has momentum, at E=0 which can't be correct. I know I have this back to front, because he also mentions that at ##E>\Delta ## that is has less momentum than but I just can't see how, could someone please explain to me what my lecture is conveying here.
 
Attachments
  • Lecture slide diagram of a classical two-state potential V(x): a central barrier of height Δ between two walls that rise to infinity, with a marked energy level E and the x axis.
    upload_2017-12-15_21-37-16.png
    37.5 KB · Views: 1,184
  • Handwritten diagram of a particle in a box with labels L, barrier and R, showing green horizontal energy levels across three regions, with the caption "any energy E>0 is valid".
    upload_2017-12-15_21-37-38.png
    44.6 KB · Views: 1,014
  • Two phase-space diagrams of a particle in a double box with a barrier: left for E < Δ with closed loops confined to the L and R boxes, right for E > Δ with a larger loop spanning both boxes through the barrier region; axes x and p, with a bracket labelled √(2mE).
    upload_2017-12-15_21-38-12.png
    48.8 KB · Views: 1,001

Discussion

Taylor_1989 said:
Here is my issue. I am very confused by the ##E<\Delta ## Beacuse I can't see how this has any momentum as it would produce an imagery number. And ye say that ##E<\Delta ## cannot be a solution yet on these diagrams he show the particle has momentum, at E=0 which can't be correct. I know I have this back to front, because he also mentions that at ##E>\Delta ## that is has less momentum than but I just can't see how, could someone please explain to me what my lecture is conveying here.
If E<Δ the particle is confined in one of the boxes, where the potential is zero.
 
This is confusing, if I look at the equation it tell me that momentum dose not exist it imaginary. I can't seem to think how this particle is moving inside the box.

I mean if I look over this again he states that at 0 poteinal the particle is in the right of left box, so the particle as I see it is moving to the left and right at the bottom of one of the boxes. But how if E=0 the ##P=\sqrt(2m(E-\Delta))## then the solution give an imaginary value so tho I can see it physically I can see how it is relating to the equation.

Is there a link to either a website or video that will help me understand this concept?

Also where is this energy barrier I can't see how this particle is moving from box to to box
 
Last edited:
Okay so I have been thinking about this a bit more. I have attched photos of my workings, and tryed to explain best to what I think is going on, I know normally it not great to post photo but I couldn’t draw a decent enough diagram to convey what I was thinking.

Handwritten sketch of a particle in a box potential: a well with energy levels E1 to E6 drawn as horizontal lines, walls at x = -x2, -x1, x1, x2 with V(x) on the vertical axis, and notes on the left box and right box below.


Handwritten derivation on lined paper of a particle-in-a-box barrier problem, with equations for energy E, momentum p = ±√(2m(E−Δ)) and the condition E > Δ, plus notes on the barrier region and the classically forbidden case.


Handwritten notes on lined paper: 'The blue line, the blue line indicate what I think the particle is doing. So by looking at my diagram I can see that if E > Δ then it will have enough momentum to leave above the boxes. And if E < Δ then it will not have enough momentum to get out of the box in the left or right. Ignoring this.'
 
Attachments
  • Handwritten sketch of a particle in a box potential: a well with energy levels E1 to E6 drawn as horizontal lines, walls at x = -x2, -x1, x1, x2 with V(x) on the vertical axis, and notes on the left box and right box below.
    25465818_10159697059840635_136173687_n.jpg
    12.6 KB · Views: 907
  • Handwritten derivation on lined paper of a particle-in-a-box barrier problem, with equations for energy E, momentum p = ±√(2m(E−Δ)) and the condition E > Δ, plus notes on the barrier region and the classically forbidden case.
    25463815_10159697060110635_409303650_n.jpg
    14.1 KB · Views: 922
  • Handwritten notes on lined paper: 'The blue line, the blue line indicate what I think the particle is doing. So by looking at my diagram I can see that if E > Δ then it will have enough momentum to leave above the boxes. And if E < Δ then it will not have enough momentum to get out of the box in the left or right. Ignoring this.'
    25463749_10159697060125635_320944506_n.jpg
    13.5 KB · Views: 903
Taylor_1989 said:
This is confusing, if I look at the equation it tell me that momentum dose not exist it imaginary. I can't seem to think how this particle is moving inside the box.

I mean if I look over this again he states that at 0 poteinal the particle is in the right of left box, so the particle as I see it is moving to the left and right at the bottom of one of the boxes. But how if E=0 the ##P=\sqrt(2m(E-\Delta))## then the solution give an imaginary value so tho I can see it physically I can see how it is relating to the equation.

Remember, the equation is p2/2m +V(x) =E
There are walls of infinite high at x1 and x4. There is a barrier between x2 and x3. The function is defined between x1 and x2, outside it is infinite.

x1<x ≤ x2 V(x) = 0
x2 <x ≤ x3 V(x) = Δ
x3 < x ≤ x4 V(x) = 0

If E < Δ the particle can not exist in the interval x2 <x ≤ x3. It is confined "inside" one of the boxes, either between x1 and x2 or between x3 and x4.

Taylor_1989 said:
Also where is this energy barrier I can't see how this particle is moving from box to to box
If E ≥ Δ the particle can exist in the whole interval x1 <x ≤ x4, only its kinetic energy changes above the barrier. It is not "inside" any boxes
 
Taylor_1989 said:
Okay so I have been thinking about this a bit more. I have attched photos of my workings, and tryed to explain best to what I think is going on, I know normally it not great to post photo but I couldn’t draw a decent enough diagram to convey what I was thinking.

View attachment 216869

View attachment 216870

View attachment 216871
The drawing is right, but I can not read your handwriting.
 
ehild said:
The drawing is right, but I can not read your handwriting.
Sorry, from your pervious comment, and a couple more hours, I realized what was going on. Thank you for the advice.
 

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