What is the purpose of trace in quantum mechanics?

[Peeter]

Reading Quantum Mechanics Demystified, and trace of an operator is covered. This includes how to compute it if one has the matrix of an operator, or how to compute it given the outer product representation of an operator.

There is however, no mention of what such a trace would be used for, and then the author moves on to expectation values.

What's the application of trace in quantum mechanics? Does zero, or real, or magnitude of, or ... (?) distinguish some physically meaningful operator characteristic?

Peeter

 

[Fredrik]

I can't think of a use for the trace of an observable like energy, but the trace operation is used a lot when dealing with density operators. See e.g. this.

 

[Meir Achuz]

Since the trace of an operator is invariant, it equals sum of the eigenvalues.

 

[Minich]

Trace is mainly used for the calculation of expectation values in the thermally equilibrium system.
For example:
Energy expectation value=Tr(H*exp(-betha*H))/Tr(exp(-betha*H))
Partition function=Tr(exp(-betha*H))
where H = hamiltonian

 

[ruleva1983]

Hi, I'm a new user so this is my first post. I say sorry in advance for my english.

As already mentioned by Fredrik and partially by Minich (in the sense that he gave an example of topic in which you use the trace), the operation of trace is very common in every field of physics in which you use the formalism of density operators. This formalism is more general than that of the original QM based on the definition of a state vector and related rules for time evolution and measurement processes. This means that the density operator formalism contains the state vector case and can describe other situations which the "simple" state vector formalism cannot describe. I'm not going to telling you the whole story of that, because it takes quite a lot of time. I'm going to say only this: there are situations in quantum mechanics in which you have a lack of information about the state vector of you system, in the sense that you know the probability that the state vector is in a state 1 and another probability that it is in a state 2. This is not the concept of superposition of states, but of mixture of states. This situations are well and completely described by the density operator formalism but cannot be described consistently with the state vector formalism.

These situations happen for example in the case mentioned by Minich in which you have an system (an atom i.e.) interacting with a reservoir at temperature T. You can only know the probability that the atom is in a certain state and this probability is connected to the temperature of the reservoir. In general these situations are common when the system A you want to study is not a closed one, but interacts with another system B (A+B being closed for example). If you are not interested in the system B and you want to look only at A, the density operator formalism, through the operation of trace, allows you to do this. So you can concentrate on the properties of system A only. There are many other things to say, I know. If you have curiosities or such, just ask...

 

[Peeter]

Thanks all! This is exactly the sort of context I figured must exist. In fact there is some coverage of density operators about 100 pages later in the text which I was able to find knowing what to look for (density operators aren't listed in the index under trace).