Unitarity Principle of Quantum Mechanics


In 1926, Max Born suggested that the probability of finding an elementary particle such as an electron in a given location is |\psi|^2. The \psi is known as the quantum mechanical wavefunction.

Quantum mechanical wavefunction can be complex-valued. When you square a complex number you can get a negative number. This is why you have to use the absolute square to get the probability density. Negative probability is not physically meaningful. Hence the absolute value signs in |\psi|^2.

If you want to find the probability of finding a particle in a specified volume then you sum up (integrate) |\psi|^2 over that volume. The sum should be equal to 1. Obviously, \psi is not God given. We construct the \psi such that \sum |\psi|^2 = 1.

The probability density interpretation of |\psi|^2 is not specific to wavefunctions over space and time. It is also valid in the case of 2-state systems such as qubits. Please see “Pure States and Mixed States in Quantum Mechanics“.

Max Born was awarded the Nobel Prize in physics for the probability density interpretation of |\psi|^2 in 1954.

What is the significance of \sum |\psi|^2 = 1?

The wavefunction evolves according to the Schrödinger or Dirac equations but while it is evolving the Born probability interpretation must be obeyed. One year from now or 1 million years from now the probability of finding this particle in this universe must still be 1. But, elementary particles do not live that long. Most of them live for a fraction of a second.

There is an enhanced (supposedly better) version of Quantum Mechanics known as the Quantum Field Theory (QFT). You can read simplistic descriptions of QFT here. In QFT the “wavefunction” is replaced by the “quantum field.” The “wavefunction” does not create or destroy particles but the “quantum field” does. If that’s the case then what does “unitarity” mean in the context of QFT? That’s a very good question and physicists do not have a good answer for that.

Well…this answer at the StackExchange is a good one but my question is still valid.

“In a quantum theory (quantum mechanics or quantum field theory), unitarity means conservation of probability (or conservation of information), that is, if a state |ψ> evolves in a state |ψ>, you will have <ψ|ψ>=<ψ|ψ>. This means that the operator which transforms |ψ> into |ψ> must be unitary. Unitarity is mandatory for the probabilistic coherence of the quantum theory.” – Trimok May 27 ’13 at 11:01

First of all, can we equate conservation of probability to conservation of information? I doubt it.  Let me know if you find a satisfactory answer.

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