Wavefunction of a 2-particle system.

[carbon9]

Hi everybody,

I'm studying Hartree-Fock theory. It is written that, in Hartree approach, for the two electrons in positions r1 and r2 (these are vectors, of course), the joint wavefunction of this 2-particle system is given as

Psi(r1, r2)=Psi1(r1) * Psi2(r2)

I'm confused a lot at this point. I have a basic knowledge of quantum mechanics and as much as I know, the wavefunction of a particle is a function of spatial coordinates (r), as in infinite square well, the wavefunction is given as

Psi(x)=sqrt(2/L) * sin(n*pi*x/L)

here, wavefunction is a function of x. It can be plotted as a function of position x and the squared modulus of wavefunction is also a function of x which gives the probability that we find it there.

Let us turn back to the 2-particle case. The wavefunction of the 2-particle system is Psi(r1, r2) that seems to be meaningless to me since there is only one coordinate system and the wavefunction had to be given something like

Psi(r)=somefunction( Psi1(r), Psi2(r)) (All Psi's are a function of independent variable r)

but it is given as

Psi(r1, r2)=Psi1(r1) * Psi2(r2)

r1 and r2 being the coordinates of particle-1 and particle-2.

Could anybody please give an explanation of this situation? We do not know r1 and r2 exactly in quantum mechanics. But how can we write the wavefunction of 2-particles as a function of r1 and r2?

[Hurkyl]

Could anybody please give an explanation of this situation?
What you have observed is an algebraic statement of what makes quantum mechanics different from classical mechanics.

[shouvikdatta8]

Dear carbon9,

I don't know anything about the Hartree-Fock theory. But, I've studied QM in my undergrad course.

First, consider what a wavefunction means. Its a function which gives info about the state of the system. For, single particle position-space wavefunctions, (like the ones you were talking about here) their modulus squared gives the probabililty of getting the particle between x and x+dx (or in 3D its r and r+dr.) Similarly, the modulus squared of 2-particle wavefunctions must give the probability of finding one particle in the range r1 & r1+dr1 AND the other particle in the range r2 & r2+dr2. In statistical lingo, AND = multiplication. So, here you just multiply 2 probabilities to get a combined probability.

I hope the above explanation will answer your question. The mathematical operation is consistent with the physical reasoning.

Regards,

Shouvik

[carbon9]

Dear Hurkyl and shouvikdatta8, thanks for your answers.

I think I've now understood something. I now think something like the picture attached to this message. Is this picture correct?

Psi(r1, r2, ..., rn) gives the combined "AND"ed probability.

Regards

[Dr Transport]

Dear Hurkyl and shouvikdatta8, thanks for your answers.

I think I've now understood something. I now think something like the picture attached to this message. Is this picture correct?

Psi(r1, r2, ..., rn) gives the combined "AND"ed probability.

Regards

Now you're getting there......

[shouvikdatta8]

Hi carbon9,

I don't see the reason why you need to consider a 3n-dimensional system. 3 coordinates are just sufficient to represent position. Also, in Quantum Mechanics you can't just pin-point positions like r1,r2.. etc. You'll have to consider an infinitesimal volume about r1,r2.. r_n, (in accordance to uncertainty requirements) so that the 'i'th particle lies between r_i and r_i + dr_i.

Regards,

Shouvik.

[carbon9]

Hi Shouvik,

Since I'm studying n-particles, the wavefunction is said to be of 3n-dimension? Is this wrong?

Regards

[Dr Transport]

Hi Shouvik,

Since I'm studying n-particles, the wavefunction is said to be of 3n-dimension? Is this wrong?

Regards

No, but to get your head around this concept, try thinking about 2 particles in 1-d then move forward.

[atyy]

Because of the problem you mentioned, for two (or more) identical particles, the wavefunction must be symmetrized/anti-symmetrized according to whether the particles are bosons are fermions. Bosons is obvious for me - you can interchange positions without changing the wavefunction. Fermions are weird, but not so weird if I remember that the wavefunction is only the probability amplitude, not the probability. Apparently there's a third sort of particle called the anyon in 2D...

[shouvikdatta8]

Observe, \Psi(r1,r2) ... since r1 and r2 are vectors, they have 3 components each and hence the wavefunction is 6-dimensional for 2 particles. However, the position space as you had considered in your diagram must be of 3-dimensions.

[carbon9]

Thanks for replies.

I think my problem lies in the misunderstanding the dimensions of functions. OK, wavefunction of 2-particles is 6-dimensional. But now consider a system of 2-particles in the real space with the following wavefunctions:

1. Psi1(r1)
2. Psi2(r2)

Then, as much as I know, the squared modulus is the probability of finding a particle at coordinate r1+dr1 and r2+dr2 for particle 1 and particle 2, respectively. OK?

But when it comes to think particle-1 AND particle-2, I still get confused. The wavefunction of 2-particle SYSTEM is

Psi(r1, r2)=Psi(x1, y1, z1, x2, y2, z2)=Psi1(x1, y1, z1)*Psi2(x2, y2, z2) is this right? I think so.

Theeen, let us come to the crucial point. What is the probability of finding particle-1 at r1+dr1 AND particle 2 at r2+dr2?

Probability density function(x1, y1, z1, x2, y2, z2)=|Psi1(x1, y1, z1)*Psi2(x2, y2, z2)|^2 ? If this is right?

Vital question is: How can we plot the wavefunction of 2-particle system?
Another: How can plot the probability density function of 2-particle system in 3-D? Since, if my expression of probability density function is correct, then it is also 6-dimensional and we can not plot it?

[atyy]

The wavefunction of 2-particle SYSTEM is

Psi(r1, r2)=Psi(x1, y1, z1, x2, y2, z2)=Psi1(x1, y1, z1)*Psi2(x2, y2, z2) is this right? I think so.

Theeen, let us come to the crucial point. What is the probability of finding particle-1 at r1+dr1 AND particle 2 at r2+dr2?

Since the particles are identical, the question doesn't make sense, as you expected. The wave function has to be symmetrized for bosons, or anti-symmetrized for fermions.
http://farside.ph.utexas.edu/teaching/qmech/lectures/node59.html
http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/IdenticalParticlesRevisited.htm

[shouvikdatta8]

You needn't worry about symmetry of wavefunctions for the time being. You CANNOT plot wavefunctions of 2-particle system in 3D because it's a function of six variables (or coordinates.) You can just imagine. . .
You are right in the other cases...

[reilly]

Think about throwing two baseballs from x=y=z=0. Throw many times in any direction you want, then you will have an event space, of ball1 at r1, and ball 2 at r2. As in(let z=0)


Ball1 Ball2
Throw

1 (5,2,0) (3,7,0)

2 (2,2,0) (5,8,0)

3 (4,9,0) (6,7,0)

and so on. There are two types of probabilities here; one for single events for ball1 and ball2, and for joint events.(This is standard probability theory.)

That is, there are probability: Prob(r1=(x1,y1,0) and Prob(r2=(x2,y2,0). Here these two single event probabilities describe independent events, and each deals with the coordinates of a single ball. These coordinates refer to the same space but to different balls; they refer to measured variables -- events as they are often called in probability theory.

Then there is the joint probability, Prob(r1=(x1,y1,0) and Prob(r2=(x2,y2,0), simultaneously, which is given here by

Prob{ r1=(x1,y1,0) x Prob(r2=(x2,y2,0)|},

a function of 6 variables, with values in the same space.

Now suppose that the tosses are not independent; after a throw of ball1, the thrower must throw ball2 in a direction + or - 30 degrees from the first throw. The product form no longer holds, which is usually the case in QM. Then, if we let the probabilities = wave function squared, the probability that ball1 is at s1 and ball2 is at s2 is given by

|Psi( r1 = s1; r2=s2|^^2.

Note this approach is constantly used in scattering theory, for both single particle detection, say the electron only in electron-proton scattering, and coincidence detection in which both particles are detected.

If you use some fancy computer graphics, like 3-d plots then you can plot multiparticle wave functions or probabilities --similar plots are often found in books dealing with special functions, with pre-computer-age plots in the form of contour maps. By hand, with a few crayons or colored pencils, you ought to be able to do your own plots -- try
psi= Sin(|r1+r2|), for example -- sorry about the lack of normalization. -- where r1 and r2 are 3- d vectors.
Regards,
Reilly Atkinson

[carbon9]

Dear atyy, shouvik and reilly,

Thanks for your answers. The links given by atyy were quite useful in understanding symmetric and antisymmetric wavefunctions and Pauli exclusion principle.

Also, I think I've understood that shouvik said, that we can not plot a 6-D function in 3D space. OK.

I thank to reilly also, especially the review of probabilty theory was a must for QM as he pointed. But, at reilly's post, I could not find a way for plotting psi= Sin(|r1+r2|) when r1 and r2 are vectors in the 3D-space. For example, I tried to plot in in MATLAB but I could not unfortunately. If there's an example plot on Internet, I'd like to look at it so, I can I think understand better.

Regards,