Schrödinger equation

In quantum mechanics, the **Schrödinger equation** is the differential equation that dictates how the wavefunction of a quantum mechanical system evolves over time.

The most general Schrödinger equation is

i
ℏ

∂

∂
t

# Ψ

H
^

Ψ

{\displaystyle i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi}

This is similar to a statement of conservation of energy; the operator on the left hand side gives the energy of the wavefunction in terms of how it changes with time, and the operator on the right hand side is the Hamiltonian operator, which gives the total energy of the system in terms of how it changes with space.

For a single particle moving at non-relativistic speeds in a scalar potential, the form of the Hamiltonian operator is simple and it can be substituted in, giving

i
ℏ

∂

∂
t

# Ψ

[

−

ℏ
2

2
μ

∇
2

V

]

Ψ

{\displaystyle i\hbar\frac{\partial}{\partial t} \Psi = \left [ \frac{-\hbar^2}{2\mu}\nabla^2 + V \right ] \Psi}

Here, μ is the mass of the particle and V is the potential energy of the particle.

## Time-Independent Schrödinger equation

Often, the wavefunction has a single defined energy value that doesn't change with time (this is known as an eigenstate), so it can be split into a component that is only dependent on space (

ψ

{\displaystyle \psi}

) and a component that is only dependent on time (

η

{\displaystyle \eta }

), such that

# Ψ
ψ
η

{\displaystyle \Psi = \psi \eta}

. Any other wavefunction can be expressed as a linear combination of the energy eigenstates, so no information is lost when taking this approach.

This gives, for a single, non-relativistic particle in a scalar potential,

1
η

i
ℏ

∂
η

∂
t

=

1
ψ

[

−

ℏ
2

2
μ

∇
2

ψ
V
ψ

]

{\displaystyle \frac{1}{\eta} i\hbar\frac{\partial \eta}{\partial t} = \frac{1}{\psi} \left [ \frac{-\hbar^2}{2\mu}\nabla^2 \psi + V \psi \right ]}

Since the left hand side depends only on time, and the right hand side depends only on space, this means that for both of them to always be equal they must be constant. This constant happens to be the energy of the system, E (called the energy eigenvalue of the eigenstate). Knowing this allows the time and space evolution of the wavefunction to be written seperately, as

i
ℏ

∂
η

∂
t

=
E
η

{\displaystyle i\hbar\frac{\partial \eta}{\partial t} = E \eta}

and

−

ℏ
2

2
μ

∇
2

# ψ

(

E
−
V

)

ψ

{\displaystyle \frac{-\hbar^2}{2\mu}\nabla^2 \psi = \left( E - V \right) \psi}

The former always has the solution

e

−

i
E
t

ℏ

{\displaystyle e^{-\frac{iEt}{\hbar}}}

, which has a modulus squared of one and therefore doesn't affect the probability amplitude, as expected. The latter is the time independent Schrödinger equation, solutions of which give you the evolution of the wavefunction over space - in other words, the probability of finding a particle at a specific location.

This equation often only has solutions at specific values of E, which is why energy is quantised.

## Specific Solutions

For some forms of the potential in the Schrödinger equation, an exact, analytic solution can be found. Some examples are derived below. Often, these exact solutions serve as good approximations to actual systems.

### Particle in a Box

A "box", as a simple quantum mechanical system, has V = 0 inside the box, and V = ∞ outside of the box, making it impossible to escape. For this example, the box is between x=0 and x=L.

Because potential is infinite outside the box, the only possible solution there is

# ψ
0

{\displaystyle \psi = 0}

. Inside of the box, the potential is 0, so the particle satisfies

−

ℏ
2

2
μ

∂
2

ψ

∂

x
2

=
E
ψ

{\displaystyle \frac{-\hbar^2}{2\mu} \frac{ \partial^2 \psi}{\partial x^2} = E \psi}

. This second order differential equation is simple; the solutions are

# ψ
A
sin
⁡

(

k
x

)

B
cos
⁡

(

k
x

)

{\displaystyle \psi = A \sin \left( kx \right) + B \cos \left( kx \right)}

, with

# k

2
E
μ

ℏ
2

{\displaystyle k = \sqrt { \frac{ 2 E \mu }{\hbar^2} }}

and A and B being two complex constants decided by boundary conditions. However, unlike a free particle, there are constraints on what this can be. Because the wavefunction must be continuous, at x=0 and x=L the value of

ψ

{\displaystyle \psi}

must be 0, as it is outside of the box. These two boundary conditions result in:

* ψ

(
0
)

=
0

{\displaystyle \psi \left( 0 \right) = 0}

, meaning that

# B
0

{\displaystyle B = 0}

.
* ψ

(
L
)

=
0

{\displaystyle \psi \left( L \right) = 0}

, meaning that

# k

n
π

L

{\displaystyle k = \frac{n \pi}{L}}

, where n is an integer.

Since k is already known in terms of energy, this puts a constraint on the possible energy levels;

# E

n
2

π
2

ℏ
2

2
μ

L
2

{\displaystyle E = n^2 \frac{\pi^2 \hbar^2}{2 \mu L^2}}

. Because n is an integer, this means that the possible energy of the particle in the box comes at discrete values; this leads to energy being quantised, the principle from which quantum mechanics was discovered.

### Quantum Harmonic Oscillator

   The potential of a classical harmonic oscillator has the form

1
2

μ

ω
2

x
2

{\displaystyle \frac{1}{2} \mu \omega^2 x^2}

, so by using this as the form of the potential for the Schrödinger equation the behaviour of a harmonic oscillator in quantum mechanics can be found. The TISE for this system is

−

ℏ
2

2
μ

∂
2

ψ

∂

x
2

=

(

E
−

1
2

μ

ω
2

x
2

)

ψ

{\displaystyle \frac{-\hbar^2}{2\mu} \frac{ \partial^2 \psi}{\partial x^2} = \left(E - \frac{1}{2} \mu \omega^2 x^2\right) \psi}

.

The derivation of the solutions to this equation is complicated and requires the use of spectral analysis, but solutions only exist for certain values of E. The first few solutions are

* ψ
0

=

1

π

b

e

−

x
2

2

b
2

{\displaystyle \psi\_0 = \sqrt{ \frac{1}{\sqrt{\pi} b} } e^{-\frac{x^2}{2b^2}} }
* ψ
1

=

2

π

b
3

x

e

−

x
2

2

b
2

{\displaystyle \psi\_1 = \sqrt{ \frac{2}{\sqrt{\pi} b^3} } x e^{-\frac{x^2}{2b^2}} }
* ψ
2

=

1

2

π

b

(

2

x
2

b
2

−
1

)

e

−

x
2

2

b
2

{\displaystyle \psi\_2 = \sqrt{ \frac{1}{2 \sqrt{\pi} b} } \left( \frac{2x^2}{b^2} - 1 \right) e^{-\frac{x^2}{2b^2}} }

Where b is a property of the harmonic oscillator defined as

# b

ℏ

μ
ω

{\displaystyle b = \sqrt{\frac{\hbar}{\mu \omega}}}

, equal to the greatest distance that the particle would achieve if it was moving under a classical potential of the same form. Each of these solutions, and higher solutions, correspond to discrete values of E, with

E
n

=

(

n

1
2

)

ℏ
ω

{\displaystyle E\_n = \left( n + \frac{1}{2} \right) \hbar \omega}

. Here, the difference between two successive energy levels is

ℏ
ω

{\displaystyle \hbar \omega}

, and even in the lowest possible energy state (called the ground state), the particle has an energy of

1
2

ℏ
ω

{\displaystyle \frac{1}{2} \hbar \omega}

.