How do you prove ##\lambda = \frac {2L} {n} ## given only L?

AI Thread Summary
The discussion revolves around proving the equation λ = 2L/n given only the distance L. The participant initially attempted to equate two equations but struggled with the constants not matching. Clarification was provided that knowing the quantum number n is unnecessary, as it is part of the kinetic energy equation. The participant later realized that their previous attempts involved de Broglie's equation and acknowledged their understanding after reviewing their notes. The conversation highlights the importance of the Schrödinger equation in solving quantum mechanics problems.
Danielk010
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Homework Statement
A free proton moves back and forth between rigid walls separated by a distance L.

If the proton is represented by a one-dimensional standing de Broglie wave with a node at each wall, show that the allowed values of the de Broglie wavelength are given by the given equation where n is a positive integer.
Relevant Equations
##\lambda = \frac {2L} {n} ##
##\lambda = \sqrt { \frac {(hc)^2} {2mc^2k} } ##, where k is the kinetic energy
## (hc)^2 = 1240 (ev *nm)^2 ##
Since I know from the equation the type of particle and the distance L, I thought of equating the first relevant equation to the second equation. Since n = 1, 2, 3 ..., I thought by equating the two equations I could get k = 1, 4, 9... and have the two constants equal each other. The two constants did not equal each other, so I am a bit stuck on where to go from here or where to start. I got an equation for kinetic energy in terms of n from my previous attempt, but I don't know the quantum number, n, nor the kinetic energy, k. Thank you for any help that can provided.
 
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Danielk010 said:
I got an equation for kinetic energy in terms of n from my previous attempt, but I don't know the quantum number, n, nor the kinetic energy, k.
What previous attempt? Did you solve the Schrodinger equation for a particle in a box? You don't need to know ##n## because it is part of ##K##. This is a "show that" kind of problem. You don't have to find any numbers.
 
kuruman said:
What previous attempt? Did you solve the Schrodinger equation for a particle in a box? You don't need to know ##n## because it is part of ##K##. This is a "show that" kind of problem. You don't have to find any numbers.
My previous attempt was trying to equate de Broglie's equation from the equation given in the question. I did not solve Schrodinger equation for a particle in a box. My class went over Schrodinger's equation involving a infinite barrier quantum well, I am not sure if that is what you are referring to. Looking back on my notes, I think I figured it out. Thank you for the help.
 
Danielk010 said:
My previous attempt was trying to equate de Broglie's equation from the equation given in the question. I did not solve Schrodinger equation for a particle in a box. My class went over Schrodinger's equation involving a infinite barrier quantum well, I am not sure if that is what you are referring to. Looking back on my notes, I think I figured it out. Thank you for the help.
Yes, I was referring to an infinite barrier well a.k.a. particle in a box. I'm glad you figured it out by yourself.
 
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