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

• I
• Higgsono
In summary: Thanks for pointing that out and giving some more concrete explanations.In summary, 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.
Higgsono
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.

Higgsono said:
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 traveling involved.

BvU said:
No it doesn't. The states are steady-state solutions and there is no traveling involved.

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

Higgsono said:
Higgsono said:
Being in a moment state means it has a certain momentum
Yes, but you mix up a classical mindset with terms used in quantum mechanics. The expectation values for the momentum are all zero for the momentum states.
Higgsono said:
I know what the mathematics says, but the physics of it I don't understand.
Stick to the math.

Higgsono said:
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 ...

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.

(warning:154 posts ! )
( 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.

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.

BvU said:
(warning:154 posts ! )
( 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 . 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.

BvU

## What is a particle in a box?

A particle in a box is a theoretical model used in quantum mechanics to study the behavior of a particle confined within a defined space, such as a one-dimensional box.

## What is the significance of calculating the number of states for a particle in a box?

Calculating the number of states for a particle in a box allows us to understand the energy levels and possible states of a confined particle, which is crucial in understanding the behavior of atoms and molecules.

## How is the number of states for a particle in a box calculated?

The number of states for a particle in a box is calculated using the Schrödinger equation, which takes into account the size of the box, the mass of the particle, and the potential energy within the box.

## What is the relationship between the size of the box and the number of states?

The number of states for a particle in a box is directly proportional to the size of the box. As the size of the box increases, the number of states also increases.

## What is the physical significance of the number of states being quantized?

The quantization of the number of states means that the particle can only exist in discrete energy levels within the box. This has significant implications in understanding the behavior of atoms and molecules, as it explains the stability of these particles and their ability to absorb and emit energy in specific amounts.

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