I Calculate the number of states for a particle in a box

The multiplicity of states for a particle in a box is proportional to the product of the volume of the box and the surface area of momentum space.

$$ \Omega = V_{volume}V_{momentum}$$

The surface area in momentum space is given by the equation:

$$p^{2}_{x}+ {p}^{2}_{y}+{p}^{2}_{z} = \frac{U}{2m}$$

But this seem rediculous to me. It says that, "the number of possible directions a particle can travel in is proportional to the total energy of the particle". I don't understand this, why would the particle have more directions to choose from when it have higher energy? I know what the mathematics says, but the physics of it I don't understand.
 

BvU

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It says that, "the number of possible directions a particle can travel in is proportional to the total energy of the particle".
No it doesn't. The states are steady-state solutions and there is no travelling involved.
 
No it doesn't. The states are steady-state solutions and there is no travelling involved.
Doesn't answer my question.

What do you mean? Being in a moment state means it has a certain momentum.
 

BvU

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BvU

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why would the particle have more directions to choose from when it have higher energy?
I can almost imagine a particle being there in that box and making is choices ... :rolleyes:

In 1D the density of states is a constant. You can imagine dots on a half-line. A line segment length ##l## starting at 0 has ##l## dots (eigenstates). Increase ##l## with ## {\sf d}l## and you get ##{\sf d}l## more eigenstates.

In 2D same - imagine dots on a grid. In a circle with radius ##r## on that grid there are ##\pi r^2## possible eigenstates. An increase of ##r## with ##{\sf d}r## gets you ##2\pi r\;{\sf d}r## more of them.

In 3D you get a sphere and with radius ##r## on that 3D grid there are ##{4\over 3}\pi r^3## possible eigenstates. An increase of ##r## with ##{\sf d}r## gets you ##4\pi r^2\;{\sf d}r## more grid points.

But you know the math.
 

BvU

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And you are definitely not the only one who wonders about this. cf @JohnnyGui here
(warning:154 posts :nb) ! )
( I hope JG doesn't mind my referring to the thread - I remember it well )

And wondering about things is a good habit for physicists !
 
Thanks for your answers! Maybe some of the confusion lies in my assumption that we are dealing with classical particles. In that case, intuition tells me that there should be a continnum of directions regardless of the energy of the particle.
 

BvU

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Well, the name Quantum Mechanics itself already suggest quantization effects ...

The transition from QM to classical (usually by means of ##\hbar\downarrow 0##) requires a delicate approach, not always intuitive. And for the particle in a box I don't even know.
 
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And you are definitely not the only one who wonders about this. cf @JohnnyGui here
(warning:154 posts :nb) ! )
( I hope JG doesn't mind my referring to the thread - I remember it well )

And wondering about things is a good habit for physicists !
I don't mind it at all :oldbiggrin:. I remember having difficulty understanding some aspects of the derivation of the continuous MB distribution, and I was too stubborn to truly nail it down in my head, hence the huge thread.
 
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