The probability of finding a particle described by the normalized state vector |Ψ> in a volume d^{3}r about r is given
by

dP(**r**,t) = |<**r**|Ψ>|^{2}d^{3}r = |Ψ(**r**,t)|^{2}d^{3}r (postulate 4).

We may interpret |<**r**|Ψ>|^{2} = |Ψ(**r**,t)|^{2} as the probability density.

|<**r**|Ψ>|^{2} = <Ψ|**r**><**r**|Ψ> 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

∫_{all space}P(**r**,t)d^{3}r = ∫_{all space} Ψ*(**r**,t)Ψ(**r**,t)d^{3}r
= <Ψ|Ψ> = 1.

We expect conservation of probability, i.e. ∂<Ψ|Ψ>/∂t = 0.

Let us verify this.

∂<Ψ|Ψ>/∂t = (∂<Ψ|/∂t )|Ψ> + <Ψ|(∂|Ψ>/∂t )

= -(iħ)^{-1}<Ψ|H|Ψ> + (iħ)^{-1}<Ψ|H|Ψ> = 0.

We use iħ∂|Ψ>/∂t = H|Ψ> and -iħ∂<Ψ|/∂t = <Ψ|H,

since H = H^{†}. Conservation of probability is thus verified.

**Is probability locally conserved?
**Local conservation of a classical quantity is usually expressed in the following way:

-(∂/∂t)∫

Here ρ(

The above equation holds for all V.

Therefore

-(∂/∂t)ρ(

[ρ(

Going from classical to quantum physics, we replace expressions for classical quantities which are functions of

ρ --> |

We then expect local conservation of probability to translate into

-(∂/∂t)<Ψ|

where <Ψ|

(<Ψ|

is the average (expectation) value of the probability current density operator.

If probability is locally conserved, then

-(∂/∂t)[Ψ*(

= (ħ/(2mi))

= (ħ/(2mi))

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 Ψ^{*}(**r**,t) and its complex conjugate by -Ψ(**r**,t) and adding the two equations.
The above expression
is then obtained. The Schroedinger equation therefore implies **local
conservation of probability.**