Conservation of Probability

## **Problem**

Consider the one-dimensional, time-dependent probability density

ρ
(
x
,
t
# )

Ψ

∗

Ψ

{\displaystyle \rho (x,t) = {\Psi}^{\*}\Psi }

and the current density

J
(
x
,
t
# )

ℏ

2
m
i

(

Ψ

∗

∂
Ψ

∂
x

−
Ψ

∂

Ψ

∗

∂
x

)

{\displaystyle J(x,t) = \frac{\hbar}{2mi} \left({\Psi}^{\*}\frac{\partial \Psi}{\partial x} - \Psi \frac{\partial {\Psi}^{\*}}{\partial x} \right)}

where

# Ψ
Ψ
(
x
,
t
)

{\displaystyle \Psi = \Psi (x,t) }

 and

Ψ

∗

=

Ψ

∗

(
x
,
t
)

{\displaystyle {\Psi}^{\*} = {\Psi}^{\*} (x,t) }

.

Show by calculation that

∂
ρ

∂
t

∂
J

∂
x

=
0

{\displaystyle \frac{\partial \rho}{\partial t} + \frac{\partial J}{\partial x} = 0 }

.

**Hint:** Use the Schrödinger equation and its complex conjugate.

i
ℏ

∂
Ψ

∂
t

=
−

ℏ
2

2
m

∂
2

Ψ

∂

x
2

V
Ψ

{\displaystyle i\hbar\frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V\Psi}

i
ℏ

∂

Ψ

∗

∂
t

=

ℏ
2

2
m

∂
2

Ψ

∗

∂

x
2

−
V

Ψ

∗

{\displaystyle i\hbar\frac{\partial {\Psi}^{\*}}{\partial t} = \frac{\hbar^2}{2m}\frac{\partial^2 {\Psi}^{\*}}{\partial x^2} - V{\Psi}^{\*} }

## **Solution**

Calculate the partial derivative of the probability density with respect to

t

{\displaystyle t}

.

∂
ρ

∂
t

=

∂

Ψ

∗

∂
t

Ψ

∂
Ψ

∂
t

Ψ

∗

{\displaystyle \frac{\partial \rho}{\partial t} = \frac{\partial {\Psi}^{\*}}{\partial t} \Psi + \frac{\partial \Psi}{\partial t} {\Psi}^{\*} }

The Schrödinger equation and its complex conjugate can be written as

∂
Ψ

∂
t

=
−

ℏ

2
m
i

∂
2

Ψ

∂

x
2

1

i
ℏ

V
Ψ

{\displaystyle \frac{\partial \Psi}{\partial t} = -\frac{\hbar}{2mi}\frac{\partial^2 \Psi}{\partial x^2} + \frac{1}{i\hbar}V\Psi}

∂

Ψ

∗

∂
t

=

ℏ

2
m
i

∂
2

Ψ

∗

∂

x
2

−

1

i
ℏ

V

Ψ

∗

{\displaystyle \frac{\partial {\Psi}^{\*}}{\partial t} = \frac{\hbar}{2mi}\frac{\partial^2 {\Psi}^{\*}}{\partial x^2} - \frac{1}{i\hbar}V{\Psi}^{\*} }

Thus

∂
ρ

∂
t

=

(

ℏ

2
m
i

∂
2

Ψ

∗

∂

x
2

−

1

i
ℏ

V

Ψ

∗

)

Ψ

(

−

ℏ

2
m
i

∂
2

Ψ

∂

x
2

1

i
ℏ

V
Ψ

)

Ψ

∗

{\displaystyle \frac{\partial \rho}{\partial t} = \left(\frac{\hbar}{2mi}\frac{\partial^2 {\Psi}^{\*}}{\partial x^2} - \frac{1}{i\hbar}V{\Psi}^{\*}\right) \Psi + \left(-\frac{\hbar}{2mi}\frac{\partial^2 \Psi}{\partial x^2} + \frac{1}{i\hbar}V\Psi\right){\Psi}^{\*} }

∂
ρ

∂
t

=

ℏ

2
m
i

(

Ψ

∂
2

Ψ

∗

∂

x
2

−

Ψ

∗

∂
2

Ψ

∂

x
2

)

{\displaystyle \frac{\partial \rho}{\partial t} = \frac{\hbar}{2mi} \left( \Psi \frac{\partial^2 {\Psi}^{\*}}{\partial x^2}-{\Psi}^{\*} \frac{\partial^2 \Psi}{\partial x^2} \right)}

Calculate the partial derivative of the current density with respect to

x

{\displaystyle x}

.

∂
J

∂
x

=

ℏ

2
m
i

(

[

∂

Ψ

∗

∂
x

∂
Ψ

∂
x

Ψ

∗

∂
2

Ψ

∂

x
2

]

−

[

∂

Ψ

∗

∂
x

∂
Ψ

∂
x

Ψ

∂
2

Ψ

∗

∂

x
2

]

)

{\displaystyle \frac{\partial J}{\partial x} = \frac{\hbar}{2mi} \left( \left[\frac{\partial {\Psi}^{\*}}{\partial x} \frac{\partial \Psi}{\partial x} + {\Psi}^{\*} \frac{\partial^2 \Psi}{\partial x^2}\right] - \left[\frac{\partial {\Psi}^{\*}}{\partial x} \frac{\partial \Psi}{\partial x} + \Psi \frac{\partial^2 {\Psi}^{\*}}{\partial x^2}\right] \right)}

∂
J

∂
x

=

ℏ

2
m
i

(

Ψ

∗

∂
2

Ψ

∂

x
2

−
Ψ

∂
2

Ψ

∗

∂

x
2

)

{\displaystyle \frac{\partial J}{\partial x} = \frac{\hbar}{2mi} \left( {\Psi}^{\*} \frac{\partial^2 \Psi}{\partial x^2} - \Psi \frac{\partial^2 {\Psi}^{\*}}{\partial x^2} \right)}

It is clear from the above calculations that

∂
J

∂
x

=
−

∂
ρ

∂
t

{\displaystyle \frac{\partial J}{\partial x} = -\frac{\partial \rho}{\partial t}}

, therefore;

∂
ρ

∂
t

∂
J

∂
x

=
0

{\displaystyle \frac{\partial \rho}{\partial t} + \frac{\partial J}{\partial x} = 0 }

## **Advanced Remark**

The conservation of probability in 3D space can be compactly expressed by the vector equation

∂
ρ

∂
t

∇

⋅
J

=
0

{\displaystyle \frac{\partial \rho}{\partial t} + \nabla \boldsymbol{\cdot J} = 0 }

where

Ψ
(
x
,
y
,
z
,
t
)

{\displaystyle \Psi (x,y,z,t) }

 and

Ψ

∗

(
x
,
y
,
z
,
t
)

{\displaystyle {\Psi}^{\*} (x,y,z,t) }

 are the wavefunction and its complex conjugate,

# ρ

Ψ

∗

Ψ

{\displaystyle \rho = {\Psi}^{\*} \Psi }

 is the probability density, and

# J

ℏ

2
m
i

(

Ψ

∗

∇
Ψ
−
Ψ
∇

Ψ

∗

)

{\displaystyle \boldsymbol{J} = \frac{\hbar}{2mi}\left({\Psi}^{\*}\nabla \Psi - \Psi \nabla {\Psi}^{\*} \right) }

 is the current density.