What is semi-classical level counting?

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In summary: However, a good resource for learning about the Hilbert-Polya conjecture and its connections to quantum mechanics is the book "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike" by Peter Borwein and Stephen Choi. This book provides a comprehensive overview of the Riemann Hypothesis and its various applications, including the connection to the Hilbert-Polya conjecture and quantum mechanics. It also includes a detailed explanation of the proof of the Gutzwiller trace formula mentioned in the conversation.
  • #1
jostpuur
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Wikipedia article Hilbert-Polya conjecture has a link to an article H=xp and the Riemann zeros by Berry & Keating. They mention that the number of energy levels below given [itex]E[/itex] could be counted by computing the area enclosed by the contour [itex]H(x,p)=E[/itex] in the phase space. What is that all about? Does there exist some theorem concerning this? Can it be justified easily? What information sources are there about this?
 
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  • #2
Yes, there is a general theorem. Consider a quantum system with [itex]d[/itex] degrees of freedom that is obtained by canonical quantization of a classical hamiltonian [itex]H(p,x)[/itex]. Suppose that the classical motion is bounded in phase space for energy [itex]E[/itex], so that the quantum states have discrete energy eigenvalues. Let [itex]\rho(E)dE[/itex] be the number of energy eigenstates with energy between [itex]E[/itex] and [itex]E+dE[/itex] (where [itex]dE[/itex] should be much larger than the mean spacing between energy eigenvalues, but small compared to any classically relevant energy scale). Then

[tex]\rho(E)=\int{d^dp\,d^dx\over(2\pi\hbar)^d}\,\delta
\bigl(E-H(p,x)\bigr)\bigl[1+O(\hbar)\big][/tex]

where [itex]\delta[/itex] is the Dirac delta function.

To derive this, first write the exact density of states as

[tex]\rho(E)=\sum_\alpha \delta(E-E_\alpha)[/tex]

where the energy eigenstates are indexed by [itex]\alpha[/itex], and [itex]E_\alpha[/itex] is an energy eigenvalue. This can be written as

[tex]\rho(E)=\sum_\alpha \langle\alpha|\delta(E-\hat H)|\alpha\rangle[/tex]

where [itex]\hat H[/itex] is the hamiltonian operator. Then we note that [itex]\sum_\alpha\langle\alpha|\ldots|\alpha\rangle[/itex] is equivalent to a trace over the Hilbert space, so we have

[tex]\rho(E)=\mathop{\rm Tr}\delta(E-\hat H)[/tex]

Then use [itex]\delta(E)=\int_{-\infty}^{+\infty}{d\tau\over2\pi\hbar}\,e^{iE\tau/\hbar}[/itex] to get

[tex]\rho(E)=\mathop{\rm Tr}\int_{-\infty}^{+\infty}{d\tau\over2\pi\hbar}\,e^{i(E-\hat H)\tau/\hbar}=\int_{-\infty}^{+\infty}{d\tau\over2\pi\hbar}\,e^{iE\tau/\hbar}\mathop{\rm Tr}e^{-i\hat H\tau/\hbar}[/tex]

Now we have the trace of the time evolution operator, which can be written as a sum over position eigenstates,

[tex]{}\mathop{\rm Tr}e^{-i\hat H\tau/\hbar}=\int d^dx\,\langle x|e^{-i\hat H\tau/\hbar}|x\rangle[/tex]

(continued next post)
 
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  • #3
Now I'm going to do some hand waving. Rigorous justification is possible but lengthy.

If the hamiltonian was of the form [itex]\hat H ={1\over 2m}\hat p^2 + V(\hat x)[/itex], and we were interested in a "short" time [itex]\tau[/itex], we could use the Campbell-Baker-Hausdorf formula to write

[tex]e^{-i\hat H\tau/\hbar} = e^{-i\hat p^2\tau/2m\hbar} \,e^{-iV(\hat x)\tau/\hbar}\,e^{O(\tau^2)}[/itex]

Then we could insert a complete set of momentum eigenstates to get

[tex]\langle x|e^{-i\hat H\tau/\hbar}|x\rangle \approx \int d^dp\,\langle x|e^{-i\hat p^2\tau/2m\hbar}|p\rangle\langle p|e^{-iV(\hat x)\tau/\hbar}|x\rangle[/itex]

Now I can remove the hats from the operators [itex]\hat p[/itex] and [itex]\hat x[/itex], because they are acting on their eigenstates, and pull these factors out front, since they are now just numbers. So we now have

[tex]\langle x|e^{-i\hat H\tau/\hbar}|x\rangle \approx \int d^dp\,e^{-i(p^2/2m+V(x))\tau/\hbar}\langle x|p\rangle\langle p|x\rangle[/itex]

Now I use [itex]\langle x|p\rangle=\langle p|x\rangle^*=e^{ipx/\hbar}/(2\pi\hbar)^{d/2}[/itex], and we have

[tex]\langle x|e^{-i\hat H\tau/\hbar}|x\rangle \approx \int {d^dp\over\,(2\pi\hbar)^d}\,e^{-iH(p,x)\tau/\hbar}[/tex]

Plugging this into our last formula for the density of states, we get

[tex]\rho(E) \approx \int_{-\infty}^{+\infty}{d\tau\over2\pi\hbar}\int{d^dp\,d^dx\over(2\pi\hbar)^d}\,e^{iE\tau/\hbar}\,e^{-iH(p,x)\tau/\hbar}[/tex]

Now carry out the integral over [itex]\tau[/itex] to get

[tex]\rho(E) \approx \int{d^dp\,d^dx\over(2\pi\hbar)^d}\,\delta
\bigl(E-H(p,x)\bigr)[/tex]

Ta da!

Of course, I cheated, because I used a small-[itex]\tau[/itex] approximation, then integrated over all [itex]\tau[/itex]. Look up the "Gutzwiller trace formula" to see how corrections to this result (which is sometimes called the "Weyl formula") are computed.
 
  • #4
Oh, I should have mentioned: the total number of states with energy less than [itex]E[/itex] is given by

[tex]N(E)=\int_0^E dE'\,\rho(E')[/tex]

Since the integral of a delta function is a step function, we have

[tex]N(E)=\int{d^dp\,d^dx\over(2\pi\hbar)^d}\,\theta
\bigl(E-H(p,x)\bigr)[/tex]

where [itex]\theta[/itex] is the step function. In one dimension, this is just the area of the [itex]x[/itex]-[itex]p[/itex] plane that is inside the contour specified by [itex]H(p,x)=E[/itex].
 
  • #5
ok. Looks great. :cool:
 
  • #6
Can anyone recommend a book about this topic?

It seems that most books about quantum theory don't include this kind of topics.
 

1. What is semi-classical level counting?

Semi-classical level counting is a method used in quantum mechanics to estimate the number of quantized energy levels in a system. It combines classical mechanics and quantum mechanics to approximate the energy levels of a system.

2. How does semi-classical level counting work?

Semi-classical level counting uses the Bohr-Sommerfeld quantization rule, which states that the action (a measure of a system's energy) is equal to an integer multiple of Planck's constant divided by the frequency of the system. This rule is then used to determine the number of allowed energy levels in the system.

3. What is the significance of semi-classical level counting?

Semi-classical level counting is important because it allows us to estimate the number of energy levels in a system without having to solve the full quantum mechanical equations, which can be complex and time-consuming. It also provides a useful approximation for understanding the behavior of quantum systems.

4. In what types of systems is semi-classical level counting commonly used?

Semi-classical level counting is often used in systems with a high degree of symmetry, such as atoms, molecules, and nuclei. It is also used in condensed matter physics to study the behavior of electrons in crystals and other materials.

5. Are there any limitations to semi-classical level counting?

Yes, semi-classical level counting is an approximation and is not always accurate, especially for systems with high energy levels. It also does not take into account quantum effects, such as tunneling, which can significantly affect the behavior of a system. Therefore, it should be used with caution and in conjunction with other methods in more complex systems.

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