Shifting of the wall problem II

[DrDu]

That seems a lot less artificial, and there's no worry about the Hilbert space, or the mass, changing underneath you.

A mathematical physicist would answer you that your limit is ill defined or that it will take a lot of work to give sense to it.

 

[stevendaryl]

A mathematical physicist would answer you that your limit is ill defined or that it will take a lot of work to give sense to it.

Really? I would think it would be less work than the approach that is being discussed.

 

[tom.stoer]

This approach with time-dependent masses seems very artificial to me.

The time-dependet mass plus the additional interaction term are consequences of the rescaling implemented as a unitary transformation between Hilbert spaces; these two formulations are strictly equivalent mathematically.

 

[tom.stoer]

I would like to present a sketch of the approach via the unitary transformation; please have a look.

We start with the scale transformation U generated by G:

$$U[\lambda] = e^{iG\lambda}$$
$$G = (px+xp)/2$$

with time-dep. lambda, such that

$$x^\prime = UxU^\dagger = x\,e^\lambda$$
$$p^\prime = UpU^\dagger = p\,e^{-\lambda}$$

results in a fixed interval

$$x^\prime \in [0,L_0]$$

The problem is equivalent to a Hamiltonian

$$H_1 = \frac{p^2}{2me^{2\lambda}}$$

with time-dep. mass plus an additional term

$$H_2 = - iU\partial_tU^\dagger = -G\dot{\lambda}$$

The ansatz for a solution of the time-dep. problem is

$$|n,t\rangle = e^{-i\int_0^t d\tau \, E_n(\tau)}\,|n\rangle$$
$$|\psi,t\rangle = \sum_n a_n(t) \, |n,t\rangle$$

with momentum eigenstates

$$p|n\rangle = k_n|n\rangle$$
$$E_n = \frac{k_n^2}{2me^{2\lambda}}$$

We have to solve the following coupled differential equations:

$$i\dot{a}_m + \dot{\lambda}\,\sum_n\langle m|G|n\rangle\,a_n = 0$$

which is formally

$$i\dot{A} = -\dot{\lambda}gA$$

with

$$A = (a_n)$$
$$g = \langle m|G|n\rangle = (k_m + k_n) \, \langle m|x|n \rangle \, / \, 2$$

The formal solution is

$$A(t) = e^{i\int_0^t d\tau \,\dot{\lambda} g}\,A_0 = e^{i(\lambda - \lambda_0) g}\,A_0$$

and therefore

$$|\psi,t\rangle = \sum_n e^{-i\int_0^t d\tau \, E_n(\tau)} \left[ e^{i(\lambda - \lambda_0) g}\,A_0 \right]_n |n\rangle$$

This basically corresponds to a kind of interaction picture with "free, time-dep." H_1 plus interaction term H_2 ~ G. The first term does not cause transitions between "instantaneous eigenstates".

The "instantaneous eigenstates" of the full, time-dep. Hamiltonian are constructed as follows

$$(H-E)|u,t\rangle = 0$$
$$|u,t\rangle = \sum_n u_n(t)|n,t\rangle$$

with eigenvalue equation for E

$$(E_m - E)u_m - \dot{\lambda}\sum_n e^{i\int_0^t d\tau\,(E_m-E_n)} \, \langle m|G|n\rangle \, u_n = 0$$

 

[DrDu]

Wonderful. The use of momentum eigenfunctions gives easy expressions for the matrix elements g.
But I have some remarks:
1. |n,t> and |-n,t> must for all t combine into a sin function which vanishes at the boundaries.
2. In the differential equation for the a's, don't you have to use the time dependent momentum functions, i.e. <m,t|G|n,t> instead of <m|G|n>? As I tried to show in my last post, the time dependent oscillations are important in preventing boundary condition violation. Namely, the elements <m|x|n> would behave something like 1/(m+n). Together with the prefactor (n+m) you can see that all elements are of order 1.

 

[tom.stoer]

Wonderful.

Thanks

|n,t> and |-n,t> must for all t combine into a sin function

Yes, you are right; this constrains the coefficients a_n and u_n.

In the differential equation for the a's, don't you have to use the time dependent momentum functions, i.e. <m,t|G|n,t> instead of <m|G|n>?

Yes, again you are right; I overlooked this. It affects the defintion of g.

=> We have to solve the following coupled differential equations:

$$i\dot{a}_m + \dot{\lambda}\,\sum_n\langle m,t|G|n,t\rangle\,a_n = 0$$

which is formally

$$i\dot{A} = -\dot{\lambda}gA$$

with

$$A = (a_n)$$
$$g = \langle m,t|G|n,t\rangle = e^{i\int_0^t d\tau\,(E_m-E_n)} \, \langle m|G|n\rangle = (k_m + k_n) \, e^{i\int_0^t d\tau\,(E_m-E_n)} \, \langle m|x|n \rangle \, / \, 2$$

This is what I can see at a first glance, but I will double check tonight.

 

[tom.stoer]

... in preventing boundary condition violation ...

Think about a related problem L²[S¹] instead of L²[0,1], i.e. the same problem but now with periodic boundary conditions. In this case something like an operator x simply does not exist, b/c u(x) = 1 ist nice but x * u(x) isn't [in our case x is not a desaster b/c x * sin() ist still OK]. So on S¹ it's harder to define the problem b/c of G ~ px+xp, am I right? How do you define x, G and the scaling on a circle?

 

[DrDu]

This looks good. Assuming n to be small and given that $E_m=\pi^2 m^2/2L^2M$ the oscillating integral will act as a cut off effectively when $E_nt\approx 1$, i.e. $k_n=\sqrt{2ME}\approx\sqrt{2M/t}$ or $\lambda=\sqrt{t/M}$ . That's the length scale where the term $x/2\, \cos( \pi x/L)$ falls off to 0 at the boundary.

 

[DrDu]

Think about a related problem L²[S¹] instead of L²[0,1], i.e. the same problem but now with periodic boundary conditions. In this case something like an operator x simply does not exist, b/c u(x) = 1 ist nice but x * u(x) isn't [in our case x is not a desaster b/c x * sin() ist still OK]. So on S¹ it's harder to define the problem b/c of G ~ px+xp, am I right? How do you define x, G and the scaling on a circle?

That's true, on a circle, phi always runs from 0 to 2π. The scaling is effected scaling the radial coordinate. Instead of scaling the mass, you scale the moment of inertia $mr^2$

 

[tom.stoer]

That's true, on a circle, phi always runs from 0 to 2π. The scaling is effected scaling the radial coordinate. Instead of scaling the mass, you scale the moment of inertia $mr^2$

What I mean is that on the circle the naive position operator x is ill-defined, and that therefore the operator xp+px is problematic, too. Have a look at the ideas in (6) in

http://arxiv.org/pdf/quant-ph/0010064.pdf

 

[tom.stoer]

One hint:

$$\exp\left[i\int_0^t d\tau\,(E_m-E_n)\right]= \exp\left[\frac{i(k_m^2 - k_n^2)}{2m}\,\int_0^t d\tau\,e^{-2\lambda}\right]$$

 

[DrDu]

What I mean is that on the circle the naive position operator x is ill-defined, and that therefore the operator xp+px is problematic, too. Have a look at the ideas in (6) in

http://arxiv.org/pdf/quant-ph/0010064.pdf

There are numerous articles which deal with how to define an approximate angle operator. The relevance is not quite clear to me, as usually you can live without this operator. As I tried to show, you also don't need an angular scaling operator.