Schrödinger Equation and Special Relativity

Question

From what I understand, the Schrödinger equation describes how the wave function of a quantum system evolves in space over a given time (I am referring to a relativistic version of the Schrödinger equation). My understanding is that the equation essentially describes the evolution of the probability of a quantum measurement as a classical system. So does this mean that the probabilities determined by the Schrödinger equation depend on the reference frame of the observer (i.e. do time dilation and length contraction affect the probabilities given by the equation)?

EDIT: What I'm ultimately wondering is if the probabilities calculated from the wave function whose evolution is described by the Schrödinger equation depend on the reference frame of the observer (i.e. if two identical systems measured by (1) someone at rest relative to the system and (2) someone in motion relative to the system, are the measurement probabilities different? Does it even make sense so say a measurement is made by someone in motion relative to the system?)

Answer 1

From what I understand, the Schrödinger equation describes how the wave function of a quantum system evolves in space over a given time (I am referring to a relativistic version of the Schrödinger equation).

First, there is no relativistic Schrodinger equation. The correct relativitic generalisation is Diracs equation, but even that is a kind of approximation to the true theory and one must work with the whole apparatus of QFT = QM + SR.

My understanding is that the equation essentially describes the evolution of the probability of a quantum measurement as a classical system.

The equation directly decribes the evolution of the probability amplitude and not probability per se; the evolution of probabilities is derived. The amplitude in some sense, is the 'square root' of probability, and is one of the basic concepts that distinguishes Quantum Mechanics from Classical Mechanics.

What I'm ultimately wondering is if the probabilities calculated from the wave function whose evolution is described by the Schrödinger equation depend on the reference frame of the observer

Yes it does. The basic problem in canonical quantisation is that we cannot make a covariant choice of creation and annihilation operators.

Answer 2

The Schrödinger equation is a non-relativistic approximation to the Klein-Gordon equation. The properties (momentum, energy, ...) described by solutions of Schrödinger equation should depend in the proper way of the Galilei reference frame. In reality they don't. The properties (momentum, energy, ...) described by solutions of Klein-Gordon equation do behave properly under Lorentz transformations, as do the solutions of the Dirac equation, which can be considered the relativistic extension of the Pauli equation.

Answer 3

In what follows I am discussing the time independent Schrodinger equation, the one we all learned in in the first quantum mechanics course, at least during my studies.

The time independent Schrodinger equation is non relativistic, and yes, it will give different solutions in different frameworks. As the wavefunctions will be different, their square, which will give the probabilities of finding the state at a given (x,y,z) in time t, will be different.

The relevant equations for relativistic situations are the Klein Gordon for bosons and the Dirac for fermions.

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles, for which parity is a symmetry, such as electrons and quarks, and is consistent with both the principles of quantum mechanics and the theory of special relativity and was the first theory to account fully for special relativity in the context of quantum mechanics.

As matter we observe is mainly composed of fermions the Dirac equations is the most relevant.

Any solution to the Dirac equation is automatically a solution to the Klein–Gordon equation, but the converse is not true.

With the formalism of quantum field theory the building blocks , solutions of the Dirac equation, are not being much discussed.

It was pointed out to me that there exist time dependent equations which are discussed in this question here, and may be of interest for people wanting to pursue this, after reading the discussion in the comments.