12,721 reputation
21334
bio website
location
age
visits member for 3 years, 10 months
seen 16 hours ago

Mar
20
comment Validity of naively computing the de Broglie wavelength of a macroscopic object
@Anna What I tried to emphasize is that the de Broglie wavelength is a property of a free degree of freedom. The internal state of the composite system is not free and will not be characterized with its constituents de Broglie wavelengths but rather with its bound state energies for example its vibrational modes.
Mar
20
comment Weyl Ordering Rule
@Ome cont. Please see also the following essay on the subject by Terence Tao: terrytao.wordpress.com/2012/10/07/…
Mar
20
comment Weyl Ordering Rule
@Ome Please see the following concise review: docs.google.com/….
Mar
20
comment Weyl Ordering Rule
@Ome The variables x and k are just dummy integration variables. The integration variables are between minus and plus infinity (This is just a Fourier transform).
Mar
19
comment How to measure a qubit in a random basis
@Niel de Beaudrap The height $z$ is uniformly distributed, $\theta$ is not uniformly distributed. The surface area of all bands $z = z_0\pm\epsilon$ is equal $4 \pi \epsilon$. Please see mathworld.wolfram.com/Zone.html. This is the Archimedes' spherical sampling theorem
Mar
10
comment How does the quantum path integral relate to the quantization of energy?
@drake Please give me a few days, I'll prepare a list.
Mar
9
comment When does $\hbar \rightarrow 0$ provide a valid transition from quantum to classcial mechanics? When and why does it fail?
@stankowait cont. Please, see the following lecture by Daniel Sternheimer on the current status of deformation quantization: guests.mpim-bonn.mpg.de/deform/dsMPIMaugust08.pdf
Mar
9
comment When does $\hbar \rightarrow 0$ provide a valid transition from quantum to classcial mechanics? When and why does it fail?
@stankowait Quantum mechanical amplitudes can be represented by expectations of products of operators. Thus, the star product allows to compute physical amplitudes. It is true that for elementary systems, other quantiation methods are more suitable. However, there are some systems such as moduli spaces and certain infinte dimensional phase spaces of physical interest, where deformation quantization is may be the only available quantization method.
Mar
3
comment Is there a Lagrangian formulation of statistical mechanics?
@Nathaniel. You are correct, the differentials in the Hamilton-Jacobi equation should be with respect to the end point. Also, the time dependence that I wrote is not the most general. In the case of an explicitely time varying Hamiltonian, the phase function depends on $t_0$ and $t_1$ and not only on their difference. In this case the time differetiation is also with respect to the end point. In fact you may choose either one of the boundary points.
Mar
1
comment Is there a Lagrangian formulation of statistical mechanics?
@Nathaniel cont. In this case instead of an iterative procedure of guessing the initial momenta and checking the second end boundary condition, we can solve the Hamilton-Jacobi equation. The price is that it is a partial differential equation. A second applications is in the discrete formulation of mechanics (we can imagine that the time "t" to be a small time step). I haven't seen an application to statistical mechanics. This is why I thought that the question is very original. But I haven't searched hard enough.
Mar
1
comment Is there a Lagrangian formulation of statistical mechanics?
@Nathaniel when $x_0$ and $x_1$ are the boundary points of a classical trajectory (satisfying the Hamilton's equations), then $S$ is the value of the classical action. Of course we can use this function for any two points on the configuration space not necessary along a classical trajectory. In this case the Hamilton Jacobi-phase function generates canonical transformations (for this reason, some authors call it the generating function). The basic application that I know of this function is in the solution of the Hamilton's equations given boundary and not initial conditions.
Mar
1
comment Is there a Lagrangian formulation of statistical mechanics?
@Vibert Spin systems can also be formulated on phase space, for example a single spin phase space can be chosen to be a 2-sphere (the collection of all spin directions). This description can also be generalized to the case of many spins, which can be formulated on "non-flat" phase spaces. The phase space is just the collection of initial data of an evolving mechanical system; The Hamiltonian contains the information about the interaction. Thus, the same description applies for both the weakly and the strongly interacting cases.
Dec
30
comment Path integral with zero energy modes
@Greg I can't see your E-mail, instead I placed the copy in a file exchange server, fileconvoy.com/… where it will be abailable in the next 7 days
Dec
28
comment Path integral with zero energy modes
@Greg, 1) The unitary transformation U does not need to preserve $\mathrm{ker}(D)$, sorry for not emphasizing that one must project on the top form after performing the action, because the top form subspace is one dimensional. I'll try to add in a few days an explicit computation of the scalar multiple and the cocycle condition. 2) I'll be happy to help if you need me to send a copy of the article.
Dec
23
comment Classical vs. Quantum use of the spin 4-vector
Sorry for the late response, I added an update explaining the ultra-relativistic case.
Dec
21
comment Classical vs. Quantum use of the spin 4-vector
The boost transformation of the spatial components of the Pauli Lubanski vector is the third equation copied from the reference article. For a massive particle, the spin is the value of the spatial component vector at its rest frame. Knowing the momentum and the Pauli-Lubanski vector, one can perform a boost to a frame where the particle is at rest and get its spin. In a frame where the particle is not at rest the spatial components of the Pauli Lubanski vector still satisfy the spin commutation relations since they are composed from the spin and an angular momentum, thus should be quantized.
Dec
20
comment Classical vs. Quantum use of the spin 4-vector
Basically, yes, but please notice that the spatial spin components in the rest frame satisfy the angular momentum commutation relations, and the x and z components cannot be measured simultinuously. Thus the numerical values of the Pauli- Lubanski vector should be understood as expectations.
Dec
11
comment Beyond WKB approximation for energies
@Emilio You may find these lecture note by B.C. Hall useful: math.cinvestav.mx/~NorteSur/Taller10/notas/Hall.pdf
Nov
20
comment Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
If I may add, if you write $a = x+ip$ and $a^{\dagger} = x-ip$, you get the harmonic oscillator Hamiltonian in the usual representation, but the harmonic osillator coordinates $x$ and $p$ are not the original coordinates that you started with.
Nov
20
comment Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
If you substitute the expressions of $a$ and $a^{\dagger}$ given in the answer in the Hamiltonian $H=\hbar\omega(a a^{\dagger}+\frac{1}{2})$ you get the Hamiltonian you started with expressed in terms of $X$ and $P$.