What Did I Do Wrong in My Quantum States Calculation?

[Saptarshi Sarkar]

Homework Statement

Consider an electron moving in 1D and confined within 0.1Å and it's speed not exceeding 10^7 m/s. The number of quantum states available to the electron is

a) 10^4
b) 56
c) 48
d) 27

Relevant Equations

n= x_0*p_0/h

I calculated the total area of phase space and divided it by the area of one cell i.e. h.

n = (x_0*m*2*v)/h

=> n = (0.1 x 10^-10 x 9.1 x 10^-31 x 2 x 10^7)/6.626 x 10^-34

=> n = 0.27

This answer doesn't match with any of the options. What did I do wrong?

Edit: The question was printed wrong, the confinement distance should be 10Å instead of 0.1. Answer comes out as 27.

 

[BvU]

Hi,
Can't understand what you are doing. Where does $n= x_0\,p_0/h$ come from ? and how does it lead to $n = x_0 \,m\,2\,v/h$ ? Does that mean $p_0 = 2mv$ ?

Do you know about particles confined to a box ?

 

[Saptarshi Sarkar]

Hi,
Can't understand what you are doing. Where does $n= x_0\,p_0/h$ come from ? and how does it lead to $n = x_0 \,m\,2\,v/h$ ? Does that mean $p_0 = 2mv$ ?

Do you know about particles confined to a box ?

This question is from Statistical Mechanics

$n= x_0\,p_0/h$ is the number of cells of area h in the phase space (momentum vs position space) of the particle. It is the no of different possible cells the particle might be in.

$n = x_0 \,m\,2\,v/h$ was used as the momentum can be both positive and negetive, so the total area of phase space doubles.

 

[BvU]

I see. Wherever the Planck constant pops up, you are in the realm of quantum mechanics. I strongly suspect your $n= x_0\,p_0/h$ is not the right expression to use...

What do you think of the expressions in the link I gave ?

 

[Saptarshi Sarkar]

I see. Wherever the Planck constant pops up, you are in the realm of quantum mechanics. I strongly suspect your $n= x_0\,p_0/h$ is not the right expression to use...

What do you think of the expressions in the link I gave ?

I know about the particle in a box problem in Quantum Mechanics. But this question is about the no of cells in the phase space which is totally a mathematical concept.

In classical systems the Phase Space is made up of points where the particle can be at and in quantum mechanical systems the phase space is divided into different microstates of equal area (volume in 3D) with the area ΔxΔp=h. So, total number of cells in 2D will be total area of phase space/h.

 

[PeroK]

I know about the particle in a box problem in Quantum Mechanics. But this question is about the no of cells in the phase space which is totally a mathematical concept.

In classical systems the Phase Space is made up of points where the particle can be at and in quantum mechanical systems the phase space is divided into different microstates of equal area (volume in 3D) with the area ΔxΔp=h. So, total number of cells in 2D will be total area of phase space/h.

The question asks for quantum states.

Homework Statement:: Consider an electron moving in 1D and confined within 0.1Å and it's speed not exceeding 10^7 m/s. The number of quantum states available to the electron is

 

[Saptarshi Sarkar]

Quantum states is what my book is calling them. I am not sure if they are the same Quantum states as Quantum eigenstates.

 

[PeroK]

Quantum states is what my book is calling them. I am not sure if they are the same Quantum states as Quantum eigenstates.

When it says (see the section I've underlined):

Homework Statement:: Consider an electron moving in 1D and confined within 0.1Å and it's speed not exceeding 10^7 m/s.

What does that mean in quantum mechanics?

 

[Saptarshi Sarkar]

When it says (see the section I've underlined):
What does that mean in quantum mechanics?

That the magnitude of the uncertainty in momentum is $10^{-31+7}$?

 

[PeroK]

That the magnitude of the uncertainty in momentum is $10^{-31+7}$?

That's not what I was thinking of. In QM the value of a dynamic quantity like speed only makes sense in the context of a measurement of that quantity - or a related quantity. In my opinion the question is poorly phrased since the electron could be in anyone of infinitly many superpositions of the allowed eigenstates.

The question is really asking for the number of possible measured values of the electron's speed below the given value. This is related to the energy eigenstates.

 

[PeroK]

Edit: The question was printed wrong, the confinement distance should be 10Å instead of 0.1. Answer comes out as 27.

I was just going to suggest this. Whatever you did, you got the same answer as the number of energy states for an infinite potential well.

 

[BvU]

Quantum states is what my book is calling them. I am not sure if they are the same Quantum states as Quantum eigenstates.

From the pdf it seems obvious that your approach was what was intended. So an error in the book answer is the most likely explanation. And @PeroK clinched it.

 

[Saptarshi Sarkar]

I was thinking about why the two answers should be same and I came up with an explanation

For a particle of mass m stuck in the x-axis from 0 to L and having maximum magnitude of momentum p

1) From phase space approach

No of microstates , $m = \frac {L2p} {h}$

2) From considering an infinite square well

Maximum energy of particle

$E_{max} = \frac {p^2} {2m} = \frac {n_{max}^2π^2h^2} {8mπ^2L^2}$

=> $n_{max} = \frac {2pL} {h}$

So, $m = n_{max}$