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

[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.

 

[kuruman]

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 Schrödinger 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.

 

[Danielk010]

What previous attempt? Did you solve the Schrödinger 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 Schrödinger equation for a particle in a box. My class went over Schrödinger'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]

My previous attempt was trying to equate de Broglie's equation from the equation given in the question. I did not solve Schrödinger equation for a particle in a box. My class went over Schrödinger'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.