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The quantum state of a system is supposed to contain all the information that can be obtained about the system such as its energy, momentum...etc.

So I have 2 questions:

1-If someone gave us a quantum state of a single particle, can we tell of what mass it is?

2-Another question is that, given a quantum state can one tell what the Hamiltonian is?

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2 - Give a state you cannot tell the Hamiltonian. It is like asking in classial mechanics given the position can you infer the Hamiltonian, simply not possible. Now if you have some data about its behaviour, Then it is possible to guess the Hamiltonian. –  Prathyush Feb 9 '13 at 7:40

2 Answers 2

1-If someone gave us a quantum state of a single particle, can we tell of what mass it is?

If you know the energy and momentum, yes, you have the mass.

mass

2-Another question is that, given a quantum state can one tell what the Hamiltonian is?

It depends on how you are given the information of the quantum state. If in mathematical functions, for example Bessel functions, then a reasonable guess might give you the Hamiltonian. If it is measurements, as in high energy physics experiments, fits need to be made to various hamiltonian guesses and probability of fits deduced. At the moment particle physics is trying to fit the Standard Model and the process seems to be converging.

If your given is an instantaneous wave function of one particle, no. One needs to accumulate the probability distribution

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Anna, you have given a nice answer. I have included the free-particle quantum state, just to make clear what one means by a quantum state and how it is presented. The relativistic analysis then follows from the quantum state. I hope you agree? I have given you +1. –  JKL Feb 9 '13 at 11:36
    
Revo, are you really two brothers sharing a name here or it is a misreading of your profile on my part? –  anna v Feb 9 '13 at 11:44

@Revo This is a very good question, and it relates to some very important properties of quantum mechanics. It is true that the quantum state of a particle contains all information we need to know about the particle, but this information is past into the particle by its Hamiltonian that contains everything the particle is and the way it interacts with its environment. Therefore, when you are talking about the quantum state you must make clear what you mean by it. A relativistic free-particle quantum state for example can be given in the form

$\psi(x) = A(p)e^{i2\pi ({\bf p.r}- Et)/h}$

From this state you can read off the momentum, $\bf p$, and the energy of the particle, $E $. From these, you can then use the relativistic analysis Anna has done in her answer to find the mass of the particle. After solving the Schrodinger or Dirac equation for the quantum state of the particle, then usually the mathematical format of the quantum states are given with as much information as possible, so that we can extract other properties of the particle we might be interested in.

If you don’t have the state in this form, and you have it in some general form, $\phi (x)$ say, then the answer is no. It is like you are given only the surname of a person and you are asked to find out who that person is. If you have some extra information about that person’s family, brothers or sisters, parents or grandparents, then it might be ok. You need to have experimental data about the scattering cross section between the particle of interest and some other particle with known mass, and from these data you can infer to the unknown mass of the particle as Anna has said.

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Of course I agree. A different pov on the question but equally valid. –  anna v Feb 9 '13 at 11:42

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