How Can We Describe a Coupled System and Heat Bath in Quantum Mechanics?

[hilbert2]

Suppose we have a quantum system with one degree of freedom, $x$, and a heat bath (possibly modeled by a large amount of coupled oscillators) with many degrees of freedom, $q_1 , q_2 , \dots , q_n$. If the system and bath are coupled by some interaction, this is a simple model of a system interacting with a measuring apparatus.

If the wavefuntion of the total system (system+bath) is of product form, $\psi (x) \phi (q_1 , q_2 , \dots , q_n )$, it seems natural to say that the wavefunction of system only is $\psi (x)$ and the wavefunction of bath only is $\phi (q_1 , q_2 , \dots , q_n )$. However, when the system and bath are interacting, entanglement between them rapidly occurs and the wavefunction is no longer of product form. (isn't this what is called "decoherence" and what happens in a measurement)

How would we assign a "wavefunction" to one small part of a larger system in the case when there is entanglement between the small part and the rest of the system? Of course there has to be some way to do this, as otherwise we would have to consider the wavefunction of whole universe every time we do QM... Do we need density matrices to do this?

 

[Jano L.]

How would we assign a "wavefunction" to one small part of a larger system in the case when there is entanglement between the small part and the rest of the system?

Entanglement (non-factorizability of $\psi$) implies that probability of x depends on the value of $q$. Of course, function of $x$ only cannot describe such correlation.

One can still seek a prescription for defining the best wave function of $x$ based on the original function $\Psi(x,q)$. For example,

$$
\psi(x) = \sqrt{\int |\Psi(x,q)|^2dq}
$$

would preserve probability distribution for $x$. However, probability distribution for momentum $p$ probably would not be preserved by this. It seems possible to enhance this by a phase factor

$$
\psi(x) = e^{i\alpha(x)}\sqrt{\int |\Psi(x,q)|^2dq}
$$

while $\alpha$ should be such function of $x$ that also probability for momenta is preserved. I do not know if such determination of $\alpha$ is mathematically tractable.

Of course there has to be some way to do this, as otherwise we would have to consider the wavefunction of whole universe every time we do QM... Do we need density matrices to do this?

Most if not all calculations with wave function assume that most of the universe can be ignored, i.e. they assume that there is wave function for few variables only. Sometimes it works! :-)

Density matrix for $x$ is much easier to derive from density matrix/wave function for $x,q$ - just sum/integrate over $q$.

Both approaches of deriving wave function and density matrix for subsystem have common problem: the resulting object obeys very complicated differential equation, so further approximations are made to simplify it.

 

[hilbert2]

Thanks for the reply. I'm somewhat familiar with the fact that the differential equation obeyed by a subsystem becomes complicated. For example, a damped harmonic oscillator can be described with a nonlinear Schrodinger equation, while in reality we would have to model the damping by coupling the oscillator to a large number of external degrees of freedom (a heat bath into which energy is dissipated).

Can you suggest any good introductory reading on the theory of decoherence/measurement?

 

[Jano L.]

Sorry I can't.

 

[kith]

How would we assign a "wavefunction" to one small part of a larger system in the case when there is entanglement between the small part and the rest of the system? Of course there has to be some way to do this, as otherwise we would have to consider the wavefunction of whole universe every time we do QM... Do we need density matrices to do this?

In order to describe an entangled subsystem you can use the so-called reduced density matrix which is obtained by tracing out all degrees of freedom which don't interest you. The resulting density matrix describes a mixed state which means that you can't assign a unique state vector. If your system is in equilibrium with the heat bath, this mixed state may be characterized by the Boltzmann distribution (or the appropriate quantum distribution).

In order to be able to assign a unique state vector to the system, you have to do state preparation. What exactly happens there depends heavily on the interpretation you are using. In textbook QM, it is simply the act of measurement which causes the system to jump into an eigenstate of the corresponding observable. In the Many Worlds Interpretation, all relevant systems including the experimenter himself become entangled, so a unique state vector can be assigned to the system relative to a certain observer state.

 

[kith]

Can you suggest any good introductory reading on the theory of decoherence/measurement?

Standard references are
http://arxiv.org/abs/quantph/0312059 (Schlosshauer 2005)
and the book "Decoherence and the Quantum-to-Classical Transition" from the same author.

I don't know how much work has been done in interpretation-independent measurement theory but if you are interested in decoherence in general, a standard reference is the book "The Theory of Open Quantum Systems" by Breuer and Petruccione