I How the probability amplitude is estimated in practice

[microsansfil]

Hi,

I would like to know how the amplitude of probability is estimated/determinated in practice, for a given experiment.

In this example 1.3.2 Analysis of Experiment 2 it is assumed that the probability for each of the two possible states are equiprobable. Than from the experimental results it is determinated the value of amplitude of probability.

In pratice, the value of the amplitudes of probabilities (which are used to calculate the probabilities of the measurement results) would therefore be determined from the experiment ?

Best Regards
Patrick

 

[DrClaude]

It is a question of state preparation: the probabilities will depends on how the quantum system was initially prepared.

For states that have the same energy, starting from a thermal source (a source that is in thermal equilibrium with some reservoir), then by the fundamental assumption of statistical mechanics, the probabilities of all states are equal.

 

[microsansfil]

It is a question of state preparation: the probabilities will depends on how the quantum system was initially prepared.
For states that have the same energy, starting from a thermal source (a source that is in thermal equilibrium with some reservoir), then by the fundamental assumption of statistical mechanics, the probabilities of all states are equal.

In the case of mixed state, where the probabilities of all state are not equal, we have to repeat the same experiment to evaluate the probabilities and then deduce the amplitude of probabilities ?

Thank
Patrick

 

[microsansfil]

http://www-thphys.physics.ox.ac.uk/people/JamesBinney/QBhome.htm

With every value in the spectrum of a given measurement there will be a quantum amplitude that we will find this value if we make the relevant measurement. Quantum mechanics is the science of how to calculate such amplitudes given the results of a sufficient number of prior measurements.

Quantum amplitudes would be deduced/built from probabilistic experiment ?

Best regards
Patrick

 

[DrClaude]

In the case of mixed state, where the probabilities of all state are not equal, we have to repeat the same experiment to evaluate the probabilities and then deduce the amplitude of probabilities ?

I should have been more clear. For a thermal source, statistical mechanics tells you that the probability of being in state of energy $E$ is proportional to $e^{-E/kT}$. So one can use knowledge of the energy states of the system to figure out the probabilities of the system to be in different (mixed) quantum states. One can also measure the ensemble before performing the experiment.

Quantum amplitudes would be deduced/built from probabilistic experiment ?

Technically, yes. In practice, it varies depending on the situation.

Lets say you are doing experiments on cold atoms, like ⁸⁷Rb. The experiment is set up such that the atoms are produced, laser-cooled, and trapped in a magneto-optical trap. After that initial stage, you perform the actual experiment you want on the atoms. You can characterize the state of the atoms at the end of the initial stage (number of atoms, velocity distribution, hyperfine state distribution, etc.) by performing this initial stage of few times, measuring these values. For a carefully done experiment, the variance will be small, such that the initial condition of the atoms used for the actual experiment is well known.