Reference: References: Robert D. Klauber, Student Friendly Quantum Field Theory, (Sandtrove Press, 2013) – Chapter 3, Problem 3.3.
In non-relativistic quantum mechanics governed by the Schrödinger equation, the probability density is given by
and the probability current is given by (generalizing our earlier result to 3-d and using natural units):
The continuity equation for probability is then
We’ll now look at how these results appear in relativistic quantum mechanics, using the Klein-Gordon equation:
We can multiply this equation by and then subtract the hermitian conjugate of the result from the original equation to get
The LHS can be written as
(use the product rule on the RHS and cancel terms).
The RHS of 6 can be written as (use the product rule again):
We can write this as a continuity equation for the Klein-Gordon equation, with the following definitions:
[The extra is introduced to make and real. Note that the factor within the parentheses in both expressions is a complex quantity minus its complex conjugate, which always gives a pure imaginary term. Thus multiplying by ensures the result is real.]
We can put this in 4-vector form if we use (for some 3-vector ):
where the implied sum over is from to (spatial coordinates), and the minus sign appears because we’ve raised the index on . If we define
(that is, the negative of 10), then . To make into a 4-vector, we add and we get
[Note that my definition of is the negative of the middle term in Klauber’s equation 3-21, although raising the index agrees with the last term in 3-21. I can’t see how his middle and last equations for and can both be right, since raising the in the middle equation for merely raises the to without changing the sign.]
The curious thing about the Klein-Gordon equation is that its probability density in 9 need not be positive, depending on the values of and its time derivative. To see how this can affect the physical meaning of the equation, consider the general plane wave solution to the Klein-Gordon equation
Klauber explores this starting with his equation 3-24, where he takes a test solution in which all and shows that so that in this case, the total probability of finding the system in some state is +1 as it should be. Let’s see what happens if we take all . In that case, 9 becomes
We now wish to calculate . We can use the orthonormality of solutions to do the integral. We have
Thus the total probability of finding the system in one of the state is negative.