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Interaction (Intermediate) Representation

December 11, 2008
From A. Messiah’s Quantum Mechanics p.321 and 722 ff. , Dover (1999)

Any unitary transformation of the vectors and the observables of the Schrödinger (or Heisenberg) representations defines a new representation.

Any problem of quantum mechanics essentially consists of the determination of the properties of the unitary operator U(t,t_0). *

In the intermediate representation, we suppose that the Hamiltonian H can be put in the form:

\displaystyle H(t)=H_0(t)+V(t)

where H_0(t) is the Hamiltonian of a Schrödinger equation whose solution is know.

Let U_0(t,t_0) be the evolution operator corresponding to H_0. We obtain vectors and observables in the interaction representation by applying the unitary tranformation U_0^{\dagger}(t,t_0) to the Schrödinger ones.

\displaystyle \left|\psi_{\mathrm{I}}(t)\right>=U_0^{\dagger}(t,t_0)\left|\psi_S(t)\right>

\displaystyle A_{\mathrm{I}}(t)=U_0^{\dagger}(t,t_0)A_S U_0(t,t_0)

The evolution operator for the states in the intermediate representation is:

\displaystyle U_{\mathrm{I}}(t, t_0)=U_0^{\dagger}(t,t_0)U(t,t_0) 

The time dependence of U_{\mathrm{I}} is determined by:

\displaystyle V_{\mathrm{I}}(t)=U_0^{\dagger}(t,t_0) V(t) U_0(t,t_0)

The vectors in the intermediate representation move in time satisfying the Schrödinger equation:

\displaystyle i\hbar\frac{\mathrm{d}}{\mathrm{d}t}\left|\psi_{\mathrm{I}}(t)\right>=V_{\mathrm{I}}(t)\left|\psi_{\mathrm{I}}(t)\right>

On the other hand, the physical quantities are represented by moving observables that are subject to the Heisenberg equation of motion written with the “unperturbed” Hamiltonian H^{0}_{\mathrm{I}}(t):

The evolution operator in the interaction representation is:

\displaystyle U_{\mathrm{I}}(t,t_0)=1-i \hbar^{-1} \int_{t_0}^t V_{\mathrm{I}}(t') U_{\mathrm{I}}(t',t_0) dt'

This Integral equation can be solved by iteration.

To fit with the introduction of the  C. Kittel’s book Quantum Theory of Solids and Abrikosov’s Methods of Quantum Field Theory in Statistical Physics, we note that, if H_0 is time independent and putting t_0=0 we have:

\displaystyle U_0(0,t)=e^{-iH_0/\hbar}

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Macroscopic Quantum Tunneling & Coherence

December 10, 2008
E. M.Chudnovsky, L. Gunther – Quantum Tunneling of Magnetization in Small Ferromagnetic Particles. Phys. Rev. Lett. 60, 661 (1988)

Macroscopic Quantum Tunneling (MQT) correspond to the tunneling of a macroscopic variable trough the barrier between two minima of the effective potential of a macroscopic system.

We consider a small ferromagnetic particle enough small to form a single magnetic domain (→ Stoner-Wohlfarth model). Equilibrium easy directions of the magnetic moment \mathbf{M} correspond to the local minima of the energy:

E=-\mathbf{M}\cdot \mathbf{H}+A_{ik}M_iM_k+B_{iklm}M_iM_kM_lM_m+\ldots

Since \mathbf{M} is an axial vector (aka pseudvector) any minimum of the energy H_0=0 is at least twice degenerate with respect to two opposite directions of \mathbf{M}.

If the exchange interaction is so much strong to suppress the dynamics of the individual spins of the particle, \mathbf{M} can be regarded as a single quantum variable.

The projection of \mathbf{M} onto one of the easy directions in general does not commute with the energy E, this means that the eigenvalues of the projection in general are not conserved quantum numbers even at H_0=0 (this is not surprising because the magnetic anisotropy appears as a result of  relativistic interactions → Landau, Lifshitz – Electrodynamics of Continuous Media).

As consequence of the last statement, \mathbf{M} can tunnel between the energy minima.

Tunneling removes the degeneracy of the ground state and put the particle into a state of lower energy wherein \langle \mathbf{M} \rangle = 0 and \langle \mathbf{M}^2 \rangle = M_0^2. Angular brackets denote quantum average.

In presence of a magnetic field, the potential energy E has, in general, one absolute minimum and several local minima so that the problem of MQT from metastable states arises.

The first reference to quantum tunneling of the magnetic moment was proposed as explanation of the experimental data indicating that transitions between different orientations of the magnetic moment in single domain nickel particles do not disappear completely with a decrease in temperature at absolute zero.

Macroscopic Quantum Coherence (experimental point of view)
For two successive measurements of \mathbf{M} separated by the time interval \Delta t one should obtain the effect of macroscopic quantum coherence:

\langle \mathbf{M}(t)\mathbf{M}(t+\Delta t) \rangle = M_0^2\cos(2P\Delta t)

at T=0, H=0 and neglecting dissipation. \hbar P is the tunneling matrix element.

For both macroscopic quantum coherence and MQT the key quantity is the tunneling rate P, which should be calculated in terms of the macroscopic parameters describing single-domain particles.

Ovrec’s Laboratory Rules

December 5, 2008
  1. Don’t take any initiative if you are alone.
  2. Perform laboratory work only when another person is present.
  3. Be quiet!

Rabi’s Oscillations

November 28, 2008

Rabi oscillations: quantum oscillations resulting from the coherent absorption and emission of photons driven by an electromagnetic wave.

The Rabi oscillations demostrate the existence of quantum superposition states in which the system is simultaneously in two or more macroscopically different quantum states!

I report the link to a good introduction to Stimulated emission and Rabi’s oscillations made by Complex Photonic Systems (COPS) group at the University of Twente. Click > here <

OECD Science, Technology and Industry Outlook 2008

November 27, 2008

In this post I report the article “Questioni di geografia” about OECD SCIENCE, TECHNOLOGY AND INDUSTRY OUTLOOK 2008 , written by Vincenzo Moretti in his Blog. Read more…

The Heisenberg Representation

November 25, 2008

\displaystyle\left|\psi_S(t)\right>=U(t,t_0)\left|\psi_S(t_0)\right>

\displaystyle\left|\psi_H\right>=U^{\dagger} (t,t_0)\left|\psi_S(t)\right>=\left|\psi_S(t_0)\right>

An observable A_S in the Schrödinger representation transforms into:

\displaystyle A_H=U^{\dagger}(t,t_0)A_SU(t,t_0)

In general A_H is not stationary even when A_S doesn’t depend explicilty upon time.

Evolution Operator

November 25, 2008

On the scale of quantum phenomena the evolution of a quantum system ceases to be strictly causal (There exist no net separation between the system itself and the observing instrument).

Postulate (1): an isolated quantum system evolves in an exactly predictable manner.

If the system was not subjected to any observation during the time interval (t_0,t), the ket vector \left|\psi(t)\right> representing the dynamical state of the system at the later time t is exactly determined by specifying \left|\psi(t_0)\right> the ket vector at the previous initial time t_0.

Postulate (2): the linear superposition of states is preserved in the course of time.

Consequently the correspondence between \left|\psi(t_0)\right> and \left|\psi(t)\right> is linear and defines a certain linear operator U(t_0,t) called the evolution operator.

\displaystyle\left|\psi(t)\right>=U(t_0,t)\left|\psi(t_0)\right>.

For a conservative system the Hamiltonian does not depend explicitly upon the time. So U(t_0,t) can be deduced by the Einstein’s postulate: E=\hbar \omega . We write the Schrödinger equation:

\displaystyle H\left| u_E(t_0)\right>=E\left| u_E(t_0)\right>

We postulate that the eigenvector \left| u_E(t)\right> evolves in time according to the law:

\displaystyle \left| u_E(t)\right>=e^{-i\hbar \omega(t-t_0)}\left| u_E(t_0)\right>

or

\displaystyle \left| u_E(t)\right>=e^{-iH(t-t_0)/\hbar}\left| u_E(t_0)\right>

Consequently:

→  \displaystyle U(t_0,t)=e^{-iH(t-t_0)/\hbar}

Differentiating we obtain:

\displaystyle i\hbar\frac{\mathrm{d}}{\mathrm{dt}} U(t,t_0)=HU(t_0,t)

satisfying the initial condition: U(t_0,t_0)=1

For a non conservative system the Hamiltonian depends explicitly upon time so the Einstein’s postulate losses all meaning. We postulate that U(t_0,t_0) is solution of the previous differential equation even when the system is not conservative, but U(t,t_0) is defined by the integral equation (Fredholm eq.):

\displaystyle U(t,t_0)=1-i \hbar^{-1} \int_{t_0}^t H(t') U(t',t_0) dt'

In order that the norm of \left|\psi (t)\right> remain constant in time, it is necessary and sufficient that H be Hermitian. If H is Hermitian, U(t_0,t_0) is an unitary operator:

UU^{\dagger}=U^{\dagger}U=1