In the algebraic formulation of Quantum Mechanics, how do probability amplitudes naturally arise?
Dear Vikas,
The following text covers the answer to your question.
The inner product between two state vectors is a complex number known as a probability amplitude. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.
For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by density matrices: self-adjoint operators of trace one on a Hilbert space. Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a positive operator valued measure. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states
For more details please use the following link:
https://en.wikipedia.org/wiki/Hilbert_space
Hoping this will be helpful,
Rafik
Dear Rafik, in the Copenhagen interpretation it seems that the term "probability" is used in an analogy with classical probability, because the observer does not know the initial microstate of the system. If not using this analogy, what would "probability amplitude" mean?
Dear Alfredo,
The meaning as invoked by Born "probability amplitudes provide a relationship between the wave function of a system and the results of observations of that system", a link first proposed by Max Born (Born probability).
Rafik
If the starting point is an algebraic formulation of Quantum Mechanics the probability amplitudes may appear somewhat mysterious. It is better to start with an abstract statistical description of preparations and measurements. From this one can define the state space as a convex set. With the extra assumption that the state space has the same shape as the set of density matrices or operators on a complex Hilbert space, all about probability amplitutedes etc. follows from certain representation theorems. Details can be found in the attached lecture notes that I wrote for a course I gave at Univ. Copenhagen on this topic.
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