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

• Danielk010
Danielk010
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.

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.

kuruman
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.

Danielk010

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