Conservation of Probability

The probability of finding a particle described by the normalized state vector |y> in a volume d3r about r is given by dP(r,t)=|<r|y>|2d3r = |y(r,t)|2d3r (postulate 4).  We may interpret |<r|y>|2= |y(r,t)|2 as the probability density.  |<r|y>|2=<y|r><r|y> is the average value of the observable |r><r| , which we may call the probability density operator.  The probability density operator is the projector onto the ket |r>.  The total probability of finding the particle at time t anywhere in space is

 .

We expect conservation of probability, i.e. .  Let us verify this.

.

We use

since H =HT.  Conservation of probability is thus verified.

Is probability locally conserved?

Local conservation of a classical quantity is usually expressed in the following way:

.

Here r(r,t) is the volume density and j(r,t)=r(r,t)v(r,t).  
The above equation holds for all V. 
Therefore 

.

[r(r,t) may stand for particle density or charge density e.t.c., and j(r,t) may stand for particle current density or electrical current density e.t.c..]

Going from classical to quantum physics, we replace expressions for classical quantities which are functions of r and p by operators which are functions of R and P and suitably symmetrize the resulting equations.

.

We then expect local conservation of probability to translate into

, 

where <y|r><r|y> is the average (expectation) value of the probability density operator and is the average (expectation) value of the probability current density operator.  If probability is locally conserved, then

,

,

= probability current density.

We can verify that this expression is a consequence of the Schroedinger equation by multiplying the Schroedinger equation in the {|r>} representation by y*(r,t) and its complex conjugate by -y(r,t) and adding the two equations.  The above expression is then obtained.  The Schroedinger equation therefore implies local conservation of probability.

The evolution operator

The Schroedinger equation is a recipe for calculating |y(t)> when |y(t0)> is known.  The Schroedinger equation is linear.  The correspondence between |y(t)> and |y(t0)> is therefore linear.  There exists a linear operator that transforms |y(t)> into |y(t0)> .

What are the properties of this evolution operator?

(a) U(t0,t0)=I.
(b) for any |y(t0)>.
Therefore .
Properties (a) and (b) completely define the evolution operator.
(c) .
Therefore We can generalize:
U(tn,t1)=U(tn,tn-1)U(tn-1,tn-2)...U(t3,t2)U(t2,t1).
.
Let t’’=t. Then , and interchanging the role of t’ and t, Therefore
(d) from the Schroedinger equation.
.
Therefore is the infinitesimal evolution operator. , since I and H are Hermitian.
, since for an infinitesimal operator we neglect terms higher than first order in dt.  The infinitesimal evolution operator is a unitary operator.  Therefore U(t,t0), which is a product of infinitesimal evolution operators, is unitary.
(e) If H does not explicitly depend on time, then we can solve for U(t,t0).  We find .

Let |En> be an eigenstate of H and let |y(t0)>=|En>.  Then is also an eigenstate . If H does not explicitly depend on time and the system is in an eigenstate of H at t=t0, then it will remain in this eigenstate.  The system is in a stationary state.