138 reputation
6
bio website math.berkeley.edu/~cawong
location Berkeley, CA
age 25
visits member for 2 years, 1 month
seen Aug 5 at 20:18

I am a fourth-year mathematics graduate student at the University of California, Berkeley.

I graduated with a B.S. in Applied and Computational Mathematics from the California Institute of Technology.

My general mathematical interests include applied functional analysis, numerical analysis, harmonic analysis, partial differential equations, numerical linear algebra, and mathematical and computational biology, chemistry, and physics.


Jul
30
comment Discreteness of set of energy eigenvalues
I agree with your physical interpretation, but I'm only commenting that it is impossible to mathematically show that the set of bound states is discrete using the Sturm-Liouville strategy, because there is no way to reduce any aspect of the problem to the S-L approach. If you try to restrict the problem to a PDE with boundary conditions (at the finite boundaries above) that are just continuity conditions, then in the weak formulation the problem fails to be self-adjoint, when you use integration by parts.
Jul
30
comment Discreteness of set of energy eigenvalues
Unfortunately it is still problematic that the interval is not finite. Even if $V(x) > E$ for sufficiently large $x$, the particle is not constrained to a finite region; it can tunnel. The way to resolve this is to abandon the Sturm-Liouville formualtion altogether. Instead, you want to look at minimax principles for the Schrodinger operator in order to find the structure of the whole spectrum, discrete and continuous, which, as you mention, will correspond to the bound and unbound states.
Jul
29
awarded  Commentator
Jul
29
comment Discreteness of set of energy eigenvalues
Unfortunately, this does not apply to the original question, because the domain is unbounded and $V \in L^{\infty}$. The resolvent of the Hamiltonian will not be compact.
Apr
15
revised Looking for modern results in semiclassical physics and relevant references
edited title
Apr
14
revised Looking for modern results in semiclassical physics and relevant references
making it conceptual
Apr
14
awarded  Editor
Apr
14
revised Looking for modern results in semiclassical physics and relevant references
rephrased the question as an actual question.
Apr
14
comment Looking for modern results in semiclassical physics and relevant references
Then my solution will be to phrase my question as an actual question.
Apr
14
comment Looking for modern results in semiclassical physics and relevant references
I find that interpretation of the function of SE to be a little strange. In my opinion, the primary role of an academic forum should be to direct people to references or papers, especially when it comes to research-level subjects as in my case.
Apr
14
asked Looking for modern results in semiclassical physics and relevant references
Apr
12
awarded  Autobiographer
Sep
21
awarded  Scholar
Sep
21
comment A good example of a nonlinear symplectomorphism?
Thanks, I really like the Hydrogen atom example. Indeed, my trouble was that every time I wrote down a Hamiltonian for some physical system I knew about, I made it too simple so that the induced symplectomorphism was linear.
Sep
21
accepted A good example of a nonlinear symplectomorphism?
Sep
21
awarded  Student
Sep
20
comment Why is torque not measured in Joules?
Torque is a vector; energy is not. They just happen to have the same units.
Sep
20
comment A good example of a nonlinear symplectomorphism?
Oh, I guess when I meant "physically useful", I meant that the coordinates were specifically used for some (hopefully easy-to-understand) physics problem. But I'll look at the Bloch equations and see if I can work out some mathematical results using them.
Sep
20
comment A good example of a nonlinear symplectomorphism?
@Qmechanic, well, I couldn't think of any good diffeomorphisms $\varphi$.
Sep
20
asked A good example of a nonlinear symplectomorphism?