Probability density that "seems" complex instead of real

[fluidistic]

Homework Statement

Hello guys,
I am extremely worried because I do not understand something. My statistical mechanics course somehow follows Reichl's book for some parts.
For a system of N particles, Reichl's define $\rho (\vec X^N, t)$ as the density of probability on the phase space. $\vec X^N$ is a point in a 6N dimensional phase space. (page 288).
A few pages later he gets that $\rho (\vec X^N, t)=e^{-i\hat L ^N t}\rho(\vec X ^N ,0)$.
Then my lecture notes go on and skipping several steps they reach that $\rho (\vec X^N, t)=\sum _{j=1}^N e^{-i \lambda _j t} c_j f_j (\vec X ^N)$ where the lambda _j's are the eigenvalues of the Liouville's operator and the f_j's are the eigenvectors of such operator. c_j's are constants (could be complex numbers I suppose).
But when I look at that expression I cannot see how rho can be real.
I took the special case of N=2 and wrote the first 2 terms of the series.

Homework Equations

Euler's equation.

The Attempt at a Solution

I believe that for rho to be real for all t, every single term of the series must be real but I'm not 100% sure.
So, by using N=2 I got the condition that $r_1 \sin (\theta _1 -t \lambda _1)=r_2 \sin (\theta _2 -t \lambda _2)$ but in my opinion that is impossible to be satisfied for all t's. Mainly because every lambda is different and so the 2 sines have a different frequency...
Anyway I also had a look a Fourier expansion of a function but in general the series starts from -j up to j in such a way that it's easy to see that all complex terms vanish. And when the series starts from j=1, everything is real. So overall there is no ambiguity that the function is real.
But in my case it's different... and I see no way how rho could be real.

Any thoughts will be appreciated. This boggles me a lot.

 

[AlephZero]

I don't know anything about statistical mechanics, but one way the expression could be real is if all the L's are pure imaginary (i.e. their real parts are zero).

That sort of thing can certainly happen in other branches of applied math. For example the "elementary" formulation of simple harmonic motion gives you the differential equation $\ddot x + \omega^2 x= 0$ where $\omega$ is the real natural frequency. But if you forget about what $\omega$ means physically and just consider it as an arbitrary differential equation with constant coefficients, you would take the eigenvalues as $\lambda_1 = i \omega$ and $\lambda_2 = -i\omega$.

Another possibility is that your function $\rho$ really is complex, but when you calculate some physically meaningful quantity, the expression will always contain something like $\rho^*\rho$, which will always be real.

 

[mfb]

Another possibility is that your function $\rho$ really is complex, but when you calculate some physically meaningful quantity, the expression will always contain something like $\rho^*\rho$, which will always be real.

I would expect this option.

 

[fluidistic]

Thanks AlephZero. Unfortunately all the lamba's are strictly real according to my lecture notes.
It's because they are the eigenvalues of a Hermitian operator (Liouville's operator) which is defined as $\hat L ^N=-i\hat H ^N$ where $\hat H ^N$ is the Hamiltonian of the N particles. (Reichl's page 292).
What I had in mind was to associate rho with |psi|^2 in QM, because both are supposed to be densities of probability and so real valued. So I'll never get $\rho^*\rho$.
Now if rho is complex valued then I completely lose faith in my lecture notes.
Because the conclusion of the notes is that a classical system will "oscillates" forever.

 

[fluidistic]

A quote from the book:

Therefore, it is useful to consider $\vec X^N$ as a stochastic variable and to introduce a probability density $\rho (\vec X^N,t)$ on the phase space, where $\rho (\vec X^N,t)d\vec X ^N$ is the probability that the state point, $\vec X ^N$, lies in the volume element $\vec X^N \rightarrow \vec X^N +d\vec X ^N$ at time t

 

[AlephZero]

Another option would be that he really means the probability is the real part of $\rho$.

Again by analogy with solving ODE's, if the coefficients in a linear ODE are real, the general form of the solution comes out naturally as a complex function, but the real and imaginary parts of it both satisfy the ODE individually. For the simple harmonic motion example again, you can take the general solution as $A \cos \omega t + B \sin \omega t$ where A and B are real, or as the real part of $C e^{i\omega t} + D e^{-i\omega t}$ where C and D are complex. Most of the math is simpler using the complex form.

 

[fluidistic]

Well I think Reichl really means density of probability and I have no problem in seeing that rho is real valued if I look at the expressions in the book. The problem arises in my lecture notes.
Both the book and my lecture notes reach the equation $\rho (\vec X ^N, t)=\exp (-it \hat L ^N ) \rho (\vec X ^N ,0)$. Here all is fine, because $\hat L ^N$ is worth $-i \hat H$ so that the exponential is real.
Here my lecture notes depart from the book. They state that $\hat L^N$ is Hermitian (the book also states this) and that $\hat L^N f_j=\lambda _j f_j$ and the fact that the operator is Hermitian implies that the lambda are real. So far I trust it.
Then it's stated that the $f_j$ form an orthogonal basis in the Hilbert space "where rho can be defined". He then writes $\rho (\vec X ^N ,0)= \sum _{j=1}^N c_j f_j(\vec X^N)$. Therefore $\hat L \rho (\vec X ^N,0)=\sum _{j=1}^N c_j \lambda _j f_j(\vec X ^N)$ which yields $\rho (\vec X^N ,t)=\sum _{j=1}^N \exp (-it \lambda _j)c_j f_j(\vec X ^N)$ which is the expression I'm having troubles with.
I am not sure where the sloppy step is.
Is it the step where he assumed that the f_j's form a basis in a Hilbert space which would imply that rho is complex valued?
Seems like the notes were dealing with classical mechanics but suddenly switched to quantum mechanics while Reichl's book stayed with classical mechanics.

 

[nrqed]

Well I think Reichl really means density of probability and I have no problem in seeing that rho is real valued if I look at the expressions in the book. The problem arises in my lecture notes.
Both the book and my lecture notes reach the equation $\rho (\vec X ^N, t)=\exp (-it \hat L ^N ) \rho (\vec X ^N ,0)$. Here all is fine, because $\hat L ^N$ is worth $-i \hat H$ so that the exponential is real.

The hamiltonian is hermitian so how can $\hat L ^N$ be hermitian? Are you sure that $\hat L ^N$ is not actually anti hermitian?

Here my lecture notes depart from the book. They state that $\hat L^N$ is Hermitian (the book also states this) and that $\hat L^N f_j=\lambda _j f_j$ and the fact that the operator is Hermitian implies that the lambda are real. So far I trust it.
Then it's stated that the $f_j$ form an orthogonal basis in the Hilbert space "where rho can be defined". He then writes $\rho (\vec X ^N ,0)= \sum _{j=1}^N c_j f_j(\vec X^N)$. Therefore $\hat L \rho (\vec X ^N,0)=\sum _{j=1}^N c_j \lambda _j f_j(\vec X ^N)$ which yields $\rho (\vec X^N ,t)=\sum _{j=1}^N \exp (-it \lambda _j)c_j f_j(\vec X ^N)$ which is the expression I'm having troubles with.
I am not sure where the sloppy step is.
Is it the step where he assumed that the f_j's form a basis in a Hilbert space which would imply that rho is complex valued?
Seems like the notes were dealing with classical mechanics but suddenly switched to quantum mechanics while Reichl's book stayed with classical mechanics.

Well, your expression $\rho (\vec X^N ,t)$ is a wave function so there is no reason it should be real. Then it does sound like the actual expression for the probability density should be the modulus squared of this.

 

[fluidistic]

The hamiltonian is hermitian so how can $\hat L ^N$ be hermitian? Are you sure that $\hat L ^N$ is not actually anti hermitian?

99% sure it's Hermitian. Not only Reichl states it at page 292, but it's given as an exercise in my course, to show that it's Hermitian. I haven't solved the problem yet, haven't really thought how to approach it yet. That's why I'm not 100% sure.

Well, your expression $\rho (\vec X^N ,t)$ is a wave function so there is no reason it should be real. Then it does sound like the actual expression for the probability density should be the modulus squared of this.

Yes. I have the feeling that my prof. initially associated rho to |psi|^2 first, just like the book does. However when my prof. departed from the book by saying that rho could be defined in a Hilbert space he slipped because the way he wrote rho, it can be associated with psi in QM and not with |psi|^2. So in a way he wasn't consistant with the definition of rho which was supposed to be a density of probability.
His conclusion that a classical system will come back to its original state is still right though, but the way he showed it isn't, I believe. At least that's the impression I get by using Reichl's book.

 

[Telemachus]

You have defined $\hat L^N=-i\hat H^N$, and you have $\rho( X^N, t)=e^{-i\hat L^N}\rho( X^N, 0)$, if you replace the definition for the Liouville operator, you get a real exponential with the Hamiltonian acting on it. It is not imaginary, the probability density decays exponentially.

Edit. I'm sorry, I see that was not the case.

 

[Telemachus]

If you have that $\displaystyle \hat H^N |\psi >=E^N | \psi >$, then: $\displaystyle e^{\hat H^N} | \psi >=e^{E^N} | \psi >$ so I think that what you must have in the exponential should be the real eigenvalues. But it is confusing, because if you think in the eigenvalues of the Liouville operator instead of the eigenvalues of the Hamiltonian it looks like you would have something complex in the exponential. If you think of it as an expansion in the hamiltonian eigenfunctions instead of the Liouville eigenfunctions, then it is clear that the exponential terms it has to be real for that expansion.

I'm not sure what happens when you express this in the basis of the Liouville operator eigenfunctions, I think that you should try working with the equality between the Hamiltonian and the Liouville operator, and trying to go from one base of eigenfuctions to the other, because it is clear to me that in the base of the Hamiltonian eigenfunction that exponential will be real naturally. Perhaps you get something real when you have the explicit form for the eigenfunctions, or the c_j's. But it is clear from the definition, or making the expansion in the basis of eigenfunctions for the Hamiltonian that rho is real.

 

[nrqed]

99% sure it's Hermitian. Not only Reichl states it at page 292, but it's given as an exercise in my course, to show that it's Hermitian. I haven't solved the problem yet, haven't really thought how to approach it yet. That's why I'm not 100% sure.

Ok, but then how do you reconcile this with the statement

$\hat L ^N=-i\hat H ^N$ where $\hat H ^N$ is the Hamiltonian ?

Yes. I have the feeling that my prof. initially associated rho to |psi|^2 first, just like the book does. However when my prof. departed from the book by saying that rho could be defined in a Hilbert space he slipped because the way he wrote rho, it can be associated with psi in QM and not with |psi|^2. So in a way he wasn't consistant with the definition of rho which was supposed to be a density of probability.
His conclusion that a classical system will come back to its original state is still right though, but the way he showed it isn't, I believe. At least that's the impression I get by using Reichl's book.

It sounds like your conclusion is correct. Hopefully you can ask him so that he can clarify the situation for the class.

 

[fluidistic]

I'm not sure what happens when you express this in the basis of the Liouville operator eigenfunctions, I think that you should try working with the equality between the Hamiltonian and the Liouville operator, and trying to go from one base of eigenfuctions to the other, because it is clear to me that in the base of the Hamiltonian eigenfunction that exponential will be real naturally. Perhaps you get something real when you have the explicit form for the eigenfunctions, or the c_j's. But it is clear from the definition, or making the expansion in the basis of eigenfunctions for the Hamiltonian that rho is real.

I do not think I'll ever get something real for all t. Because the only dependence on t is in the exponentials. The eigenfunctions of the Liouville's operator (at least as written in my notes) do not depend on time.

Ok, but then how do you reconcile this with the statement

$\hat L ^N=-i\hat H ^N$ where $\hat H ^N$ is the Hamiltonian ?

I've no idea. Apparently here's a proof of the Hermiticity: http://www.nyu.edu/classes/tuckerman/mol.dyn/lectures/lecture_9/node2.html.

It sounds like your conclusion is correct. Hopefully you can ask him so that he can clarify the situation for the class.

Yeah I will try...

 

[Telemachus]

But by definition $\rho$ was a real valued function, and that should hold for the series expansion. I've studied from Reichl for Statistical Mechanics too, and I followed most of the deductions, and examples. I didn't noted this before, but what I can tell you is that I've found some mistakes in the book. I can't tell you right now where are those mistakes because I gave my notes to a friend of mine, it was silly mistakes anyway, but its a possibility that there is a mistake in the definition of the Liouville operator. As nrqed showed, it doesn't look like an hermitian operator, because H is hermitian, and when you multiply it by i, you get complex numbers in the diagonal, right?

 

[fluidistic]

Yeah now that I look more into the book I'm getting more confused. I think I'll just asume for now that the probability density of a classical system will oscillate in time due to the Liouville's equation.

 

[nrqed]

I do not think I'll ever get something real for all t. Because the only dependence on t is in the exponentials. The eigenfunctions of the Liouville's operator (at least as written in my notes) do not depend on time.


I've no idea. Apparently here's a proof of the Hermiticity: http://www.nyu.edu/classes/tuckerman/mol.dyn/lectures/lecture_9/node2.html.

Ok, thanks. Now I see what this L is. But then it cannot be equal to i times the Hamiltonian.
Where did you see this identification ($L =- i H$)? It is not in that derivation. Is it in the book? In the class notes?

 

[fluidistic]

Ok, thanks. Now I see what this L is. But then it cannot be equal to i times the Hamiltonian.
Where did you see this identification ($L =- i H$)? It is not in that derivation. Is it in the book? In the class notes?

Both in my lecture notes and the book (page 292).Edit: Hmm actually he uses $\hat H^N$ and then a curved hat H and it's extremely confusing to me what he's doing/defining on page 291.
He speaks about a Poisson bracket for what seems to be a sum of Poisson brackets and to me it seems he mixes H for the curved H.

 

[nrqed]

Both in my lecture notes and the book (page 292).


Edit: Hmm actually he uses $\hat H^N$ and then a curved hat H and it's extremely confusing to me what he's doing/defining on page 291.
He speaks about a Poisson bracket for what seems to be a sum of Poisson brackets and to me it seems he mixes H for the curved H.

Hi again,
I managed to get my hands on a copy of Reichl. But on pages 291 and 292, he is doing classical mechanics, not quantum mechanics. His H is the operator that implements the Poisson bracket. Can you tell me where in the book he generalizes these results to quantum mechanics?

 

[fluidistic]

Hi again,
I managed to get my hands on a copy of Reichl. But on pages 291 and 292, he is doing classical mechanics, not quantum mechanics. His H is the operator that implements the Poisson bracket. Can you tell me where in the book he generalizes these results to quantum mechanics?

Ok nice.
Well the part of the book I was reading never generalized this to quantum mechanics (as I said at the end of post #7), he stayed with Classical Mechanics.

I had a feeling I could make an analogy with Reichl's $\rho(\vec X^N , t)$ to the quantum mechanics $|\Psi (x,t)|^2$ because both are supposed to be densities of probability and therefore real.
However the way my prof. wrote down $\rho(\vec X^N , t)$, it seems it could be associated to $\Psi(x,t)$ instead of its modulus squared, right after he said that he could define rho into a Hilbert space.

 

[fluidistic]

No wonder I was confused on page 291 about H and the curved H (http://librarum.org/book/10896/309 ).
The book I own in my hands is a different edition (1st, year 1980) and that page is page 192 and he's consistantly using the curved H, unlike the newer version which mixes both H confusingly to me!

Anyway if the operator L is Hermitian, it must have real eigenvalues and even what Reichl gets seems complex valued now to me: $\rho (\vec X^N, t)=e^{-i\hat L ^N t}\rho(\vec X ^N ,0)$. You might tell me to replace L by -iH but I won't do that. I'll keep it as the book did. The argument used by the book regarding this expression is that it oscillates in time because L is Hermitian and has real eigenvalues. He is clearly saying that the probability density will oscillate in time unlike the Fokker-Planck equation.

So I am asking, as a student, is rho really a density of probability?

 

[Telemachus]

It is, look at the definition that she gives in the last paragraph of page 287. I think that rho has to be real, you can get the oscillatory behavior by taking the real or imaginary part of it.

(It's she, Linda E. Reichl btw).

 

[fluidistic]

It is, look at the definition that she gives in the last paragraph of page 287. I think that rho has to be real, you can get the oscillatory behavior by taking the real or imaginary part of it.

(It's she, Linda E. Reichl btw).

Page 288 you mean?
She defines rho as a probability density and then she writes $P(R)=\int _R \rho (\vec X ^N ,t)d \vec X ^N$ where P(R) is the probability to find the system in the region R. It's clearly real as well as rho.

But the way she manipulated it and reached $\rho (\vec X^N, t)=e^{-i\hat L ^N t}\rho(\vec X ^N ,0)$ cannot be right. Or I'm missing something?

If, as you say, rho is real, why are you taking the real and imaginary parts of it to get the oscillatory motion?

 

[Telemachus]

Well, you can do that when you work with a Fourier series, you can use imaginary exponentials to express a real function, right?

 

[nrqed]

Page 288 you mean?
She defines rho as a probability density and then she writes $P(R)=\int _R \rho (\vec X ^N ,t)d \vec X ^N$ where P(R) is the probability to find the system in the region R. It's clearly real as well as rho.

But the way she manipulated it and reached $\rho (\vec X^N, t)=e^{-i\hat L ^N t}\rho(\vec X ^N ,0)$ cannot be right. Or I'm missing something?

If, as you say, rho is real, why are you taking the real and imaginary parts of it to get the oscillatory motion?

I finally had a minute to have a closer look. I am looking at the second edition.
She defines
$L = - i H$ so $e^{-i L } = e^{-H}$ which is real because the operator $H$ is real. So everything is real here, there are no problems.

I think it is simply wrong to simply say that this can be directly promoted to a quantum theory verbatim. One could say that it $looks$ like the time evolution of a wave function with hamiltonian $\hat{H} = \hat{L}$ but as you have been saying from the beginning, the wave function would now have to be $\rho (X,t)$ which does not fit with its interpretation, so it simply does not seem like a meaningful thing to do. If one insists on doing this, one must then give a precise interpretation of this quantum [iex] \rho [itex] which, I gather, was not done by the prof (or did he/she??)

 

[fluidistic]

Well, you can do that when you work with a Fourier series, you can use imaginary exponentials to express a real function, right?

Well there's a difference. If you write a function as a sum of complex exponentials the sum goes from -N to N in which case it's trivial to see that every single complex term gets canceled (just use Euler's notation and you'll see that every sine cancels out). If you use the notation of sum of cosines and sines from i=1 to N, then every coefficient in the series is real. Overall there is no ambiguity that the function you get is real.
But in Reichl's book and in my lecture notes it's a totally different story.

I finally had a minute to have a closer look. I am looking at the second edition.
She defines
$L = - i H$ so $e^{-i L } = e^{-H}$ which is real because the operator $H$ is real. So everything is real here, there are no problems.

I fully agree that the problem gets removed once you replace L in terms of H.
But if you follow her arguments, the eigenvalues of L are real, so that you can replace L by a real number in the exponential and you get that oscillatory motion that she mentions.

So if we replace L by -iH we don't get a rho that oscillates, which goes against the conclusion of that section. And it would mean that, assuming the eigenvalues of H are real and positive, that rho diverges with time. If the eigenvalue(s) is/are negative, the system goes toward equilibrium and Reichl would be happy and there would be no need to deal with Fokker-Planck equation and probably all what follows.

Basically she states that with the Newton's equations alone, a system won't reach equilibrium due to rho being oscillatory. So that she has to go deeper and use another theory/ies.

I think it is simply wrong to simply say that this can be directly promoted to a quantum theory verbatim. One could say that it $looks$ like the time evolution of a wave function with hamiltonian $\hat{H} = \hat{L}$ but as you have been saying from the beginning, the wave function would now have to be $\rho (X,t)$ which does not fit with its interpretation, so it simply does not seem like a meaningful thing to do. If one insists on doing this, one must then give a precise interpretation of this quantum [iex] \rho [itex] which, I gather, was not done by the prof (or did he/she??)

I don't know yet, we've just reached that part although I have a class in about 1 hour from now.Edit: I may be wrong that it's trivial to see that every complex term in a Fourier series cancels out. But they have to anyway, because the sum has to be real.
And taking the real part of a real function should leave it intact.

 

[Telemachus]

I finally had a minute to have a closer look. I am looking at the second edition.
She defines
$L = - i H$ so $e^{-i L } = e^{-H}$ which is real because the operator $H$ is real. So everything is real here, there are no problems.

I think it is simply wrong to simply say that this can be directly promoted to a quantum theory verbatim. One could say that it $looks$ like the time evolution of a wave function with hamiltonian $\hat{H} = \hat{L}$ but as you have been saying from the beginning, the wave function would now have to be $\rho (X,t)$ which does not fit with its interpretation, so it simply does not seem like a meaningful thing to do. If one insists on doing this, one must then give a precise interpretation of this quantum [iex] \rho [itex] which, I gather, was not done by the prof (or did he/she??)

I think that the quantum counterpart of this operator is the probability density operator that Reichl defines later in that book. This one: http://en.wikipedia.org/wiki/Density_matrix

If you ask to your professor let us know the answer, I'm interested to know it :D

BTW, the poisson bracket that Reichl defines as $\hat H$ is it Hermitian? because everybody here thought of the Hamiltonian at that time, but now I'm not sure that it is hermitian (the Hamiltonian, which is the quantum counterpart of that operator is Hermitian, but I'm not sure that it implies that the classical poisson bracket is Hermitian too). The thing is I don't think that if you multiply an hermitian operator by i you can get another hermitian operator. I'm thinking the operator in a matrix representation, if you multiply by i you would get complex numbers in the diagonal, and it couldn't be hermitian, right?

 

[fluidistic]

Thanks for the insight Telemachus. I did not understand the explanation of my prof today, so I asked to a tutor and he gave me some ideas.
I don't know whether the following is right but from $\rho (\vec X ^N ,t)=e^{-i \hat L t } \rho (\vec X ^N ,0)$, I rewrote $\rho (\vec X ^N ,0)$ as $\sum _{j=1}^N c_j \psi _j (\vec X ^N)$.
So that $\rho (\vec X ^N, t)= e^{-i \hat L t } \sum _{j=1}^N c_j \psi _j (\vec X ^N)$.
And this is equal to $$\sum _{j=1}^N c_j e^{-i \lambda _j t} \psi _j (\vec X ^N)$$ where I assumed that the lambda_j's are the eigenvalues associated to the psi_j's eigenfunctions for the Liouville's operator. I assume these eigenvalues are real.
Wait... hmm that's exactly what my prof had done. Well, now that I think about it, if the psi are not necessarily real functions then maybe it's possible to obtain a real $\rho (\vec X ^N ,t)$ that oscillates with time, as desired? I think that all the psi's might have to get a complex part for this to happen though, I am not 100% sure. And if this is true then the c_j might also be complex valued so that overall it would be possible to build a real valued and oscillatory rho of X, t.

Now the only thing to check is whether it's possible that the eigenfunctions of the Liouville operator can be complex valued (I think it's quite possible!). But of course the eigenfunctions of that operator depend on the system studied...
Anyway I'll take this as the solution to the mystery. Rho(X,t) is indeed real, the Liouville operator is indeed Hermitian (to prove it, you can show that <L psi , phi > = <psi , L phi> where the inner product consists of 6N integrals. And you use the fact that L=-iH with the H given in the book that contains partial derivatives. After some algebra and assuming that the integral of some sums are equal to the sum of some integrals you'll have to integrate by parts and the equality should pop up so that it is really Hermitian).

 

[ChrisVer]

And why do you, so desperately, want a real probability density?
It's not probability in the first place.
Am I wrong?

 

[Telemachus]

Otherwise you would have an imaginary probability when you integrate it. Even in quantum mechanics the probability density is given by a real function (the square modulus of the wave function, which is complex).

Actually, the probability density as here defined is a probability per unit volume in phase space. So in essence, it is a probability.

 

[ChrisVer]

Ah sorry- yes I made a mistake with the language... I thought you had to take $| \rho |^{2}$ then and integrate.