Finding Probability of Two-Identical-Particle System in a Given State

[Samama Fahim]

Problem: A system contains two identical spinless particles. The one particle states are spanned by an orthonormal system $|\phi_k>$. Suppose that particle states are $|\phi_i>$ and $|\phi_j>$ ($i \neq j$). (a) Find the probability of finding the particle in the state $|\xi>$ and $|\eta>$ (not necessarily eigenstate) (b) What is the probability that one of them is in state $|\xi>$? (c) Suppose now that the particles are not identical and they are measured with an instrument that cannot distinguish between them. Given an answer to part (a) and (b) in this case.

If the two identical particles are fermions, then we might probably write:

$$\Psi_{i,j}(\vec{r_1},\vec{r_2}) = \frac{1}{\sqrt{2}}(\phi_i(\vec{r_1}) \phi_j(\vec{r_2}) - \phi_i(\vec{r_2}) \phi_j(\vec{r_1})).$$

And if these particles are bosons, we might write:

$$\Psi_{i,j}(\vec{r_1},\vec{r_2}) = \frac{1}{\sqrt{2}}(\phi_i(\vec{r_1}) \phi_j(\vec{r_2}) + \phi_i(\vec{r_2}) \phi_j(\vec{r_1})).$$

Do symmetric or anti-symmetric wavefunctions form a complete set? Can we write some wave function $\Phi$ as

$$\Phi = \sum_{i,j} c_{ij} \Psi_{i,j}$$?

Perhaps, one of the $\Psi_{i,j}$ equals $\Psi_{i,j}(\vec{r_1},\vec{r_2}) = \frac{1}{\sqrt{2}}(\xi(\vec{r_1}) \eta(\vec{r_2}) \pm \eta(\vec{r_2}) \xi(\vec{r_1}))$ and we are supposed to find the inner product of $|\xi, \eta>$ with $\Phi$? Is that what the problem asks one to do? If not, please give me a hint as to what the problem is really about. If you would know the source of the problem, please refer me to it.

 

[PeterDonis]

Is that what the problem asks one to do? If not, please give me a hint as to what the problem is really about.

Is this a homework problem?

If you would know the source of the problem, please refer me to it.

I'm confused about why you don't know the source of the problem. How can you state the problem if you don't know where it came from?

 

[Samama Fahim]

Is this a homework problem?I'm confused about why you don't know the source of the problem. How can you state the problem if you don't know where it came from?

It's been assigned by the instructor.

 

[Samama Fahim]

Is this a homework problem?I'm confused about why you don't know the source of the problem. How can you state the problem if you don't know where it came from?

Yes it is a homework problem. Could you give me a hint where I might start?

 

[PeterDonis]

Yes it is a homework problem.

Exactly what part of the OP of this thread is the problem, and what part is your attempts at a solution?

 

[Samama Fahim]

Exactly what part of the OP of this thread is the problem, and what part is your attempts at a solution?

What are $|\xi>$ and $|\eta>$? Are these single particle states? I don't know where to start since I don't understand the problem statement. The only thing I know is how to write symmetric and antisymmetric wave function. What follows the problem statement in the OP is my attempt.

 

[PeterDonis]

I don't know where to start since I don't understand the problem statement.

What part of the OP is the problem statement? Where does the problem statement end and your questions about it start?

 

[PeterDonis]

I don't know where to start since I don't understand the problem statement.

Are you sure you're giving us the problem statement exactly as it was given to you? If you have some way to link to the actual problem statement, or post an image of it, that would help.

 

[Samama Fahim]

Problem: A system contains two identical spinless particles. The one particle states are spanned by an orthonormal system $|\phi_k>$. Suppose that particle states are $|\phi_i>$ and $|\phi_j>$ ($i \neq j$). (a) Find the probability of finding the particle in the state $|\xi>$ and $|\eta>$ (not necessarily eigenstate) (b) What is the probability that one of them is in state $|\xi>$? (c) Suppose now that the particles are not identical and they are measured with an instrument that cannot distinguish between them. Given an answer to part (a) and (b) in this case.

If the two identical particles are fermions, then we might probably write:

$$\Psi_{i,j}(\vec{r_1},\vec{r_2}) = \frac{1}{\sqrt{2}}(\phi_i(\vec{r_1}) \phi_j(\vec{r_2}) - \phi_i(\vec{r_2}) \phi_j(\vec{r_1})).$$

And if these particles are bosons, we might write:

$$\Psi_{i,j}(\vec{r_1},\vec{r_2}) = \frac{1}{\sqrt{2}}(\phi_i(\vec{r_1}) \phi_j(\vec{r_2}) + \phi_i(\vec{r_2}) \phi_j(\vec{r_1})).$$

Do symmetric or anti-symmetric wavefunctions form a complete set? Can we write some wave function $\Phi$ as

$$\Phi = \sum_{i,j} c_{ij} \Psi_{i,j}$$?

Perhaps, one of the $\Psi_{i,j}$ equals $\Psi_{i,j}(\vec{r_1},\vec{r_2}) = \frac{1}{\sqrt{2}}(\xi(\vec{r_1}) \eta(\vec{r_2}) \pm \eta(\vec{r_2}) \xi(\vec{r_1}))$ and we are supposed to find the inner product of $|\xi, \eta>$ with $\Phi$? Is that what the problem asks one to do? If not, please give me a hint as to what the problem is really about. If you would know the source of the problem, please refer me to it.

Problem statement ends here "Given an answer to part (a) and (b) in this case." What follows is my attempt.

 

[Samama Fahim]

Are you sure you're giving us the problem statement exactly as it was given to you? If you have some way to link to the actual problem statement, or post an image of it, that would help.

I tried googling the problem statement but nothing relevant came up. Not even close.

 

[PeterDonis]

Problem statement ends here "Given an answer to part (a) and (b) in this case." What follows is my attempt.

Ok, thanks, that's helpful.

My first observation would be that, since the problem statement says the particles are spinless, you know they are bosons, so that's the only possibility you need to consider for the case where they are identical/indistinguishable.

What are $|\xi>$ and $|\eta>$? Are these single particle states?

I think that is the intention; that is, I think part (a) is asking for the probability of finding one particle in state $\ket{\xi}$ and one particle in state $\ket{\eta}$, and part (b) is asking for the probability of finding one particle in state $\ket{\xi}$.

However, the best way to know for sure what the intent of the problem statement is is to ask your instructor about anything you are not sure about. We can only speculate here because we didn't assign the problem, your instructor did.

 

[PeterDonis]

if these particles are bosons, we might write:

$$\Psi_{i,j}(\vec{r_1},\vec{r_2}) = \frac{1}{\sqrt{2}}(\phi_i(\vec{r_1}) \phi_j(\vec{r_2}) + \phi_i(\vec{r_2}) \phi_j(\vec{r_1})).$$

This is not the only possible wave function for a state with two indistinguishable bosons. You also have to account for the possibility of both bosons being in the same state.

 

[Samama Fahim]

Ok, thanks, that's helpful.

My first observation would be that, since the problem statement says the particles are spinless, you know they are bosons, so that's the only possibility you need to consider for the case where they are identical/indistinguishable.I think that is the intention; that is, I think part (a) is asking for the probability of finding one particle in state $\ket{\xi}$ and one particle in state $\ket{\eta}$, and part (b) is asking for the probability of finding one particle in state $\ket{\xi}$.

However, the best way to know for sure what the intent of the problem statement is is to ask your instructor about anything you are not sure about. We can only speculate here because we didn't assign the problem, your instructor did.

Bosons also have spin. Don't they?

 

[Samama Fahim]

This is not the only possible wave function for a state with two indistinguishable bosons. You also have to account for the possibility of both bosons being in the same state.

Do symmetric wave functions of this sort form a complete set?

 

[PeterDonis]

Bosons also have spin. Don't they?

Spin-zero, i.e., spinless, particles are bosons. Zero is an integer, and any particle with integer spin is a boson.

Do symmetric wave functions of this sort form a complete set?

They do in the sense that any symmetric wave function can be expressed as a linear combination of a basis set of symmetric wave functions.

 

[PeterDonis]

You also have to account for the possibility of both bosons being in the same state.

Reading the problem statement again, it might rule this out. If the problem statement specifies that the two bosons start out in different one-particle states, then the wave function for the two-particle system at the start would only need to account for that possibility.

 

[PeterDonis]

Do symmetric wave functions of this sort form a complete set?

I don't know that this is relevant to the problem. If you know the wave function of the two-particle system at the start, then computing probabilities of possible measurement results is just a matter of taking the inner product of that wave function with the appropriate wave functions describing those measurement results.