Schrodinger's Wave Equation - Solutions, Derivation and Time Dependent and Independent Equation

Developed and formulated by the Austrian physicist Erwin Schrödinger in 1926, Schrodinger’s equation describes the energy and position of an electron in space and time while taking into account the wave matter duality nature of that electron inside the atom. Schrodinger’s wave equation, or just Schrodinger’s equation as its commonly referred to, is analogous to Newton’s laws of motion. If the initial conditions are known, the possible positions of the subatomic particles measured can be predicted. However, Schrodinger’s wave equation is not as deterministic in nature as is the case with Newton’s law of motion. This is because by knowing the initial position of the object and the number of forces acting on the object, the position of the object after any finite length of time can be determined using Newton’s laws of motion. Whereas Schrodinger’s wave equation cannot specify the exact location of the particles, but rather only the possible positions and the probability percentage of the particle being in a particular position. Thus, owing to this reason, Schrodinger’s wave equation is deterministic only at the statistical level and not at the individual particle level. Schrodinger came up with the wave equations by conducting extensive studies and experiments on hydrogen atoms, giving him conclusions with utmost accuracy. The position-space Schrodinger equation for a single non-relativistic particle in one dimension is given as follows.

i ħ. (d Ψ / dt) (x, t) = [- ħ²/ 2m (d²/ dx²) + V (x, t)] Ψ (x, t)

where

• Ψ (x, t) is a wave function that assigns a complex number to each point in space x at each time t
• m is the mass of the measured particle
• V (x, t) is the potential that represents the environment in which the particle exists
• i is the imaginary unit constant
• ħ is the reduced Planck’s constant

Based on the dependence of the wave equation with respect to time, there are two different forms of Schrodinger’s wave equation, namely the time-independent Schrodinger wave equation and the time-dependent Schrodinger wave equation.

Time-Dependent Schrodinger's Wave Equation

The time-dependent form of the Schrodinger wave equation describes the system which is evolving with time, and so it depends on the physical situation of that system. The expression for the time-dependent form of Schrodinger’s wave equation is given as follows.

i ħ d (Ψ (t))/ dt = Ĥ ((Ψ (t))

where

• Ψ is the state vector of the quantum system
• t is time
• Ĥ is an observable, Hamiltonian operator.

The Hamiltonian of a system is an operator that corresponds to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of all possible outcomes that can be obtained when the measurement of the system's total energy is done. The derivation for the time-dependent form of the Schrodinger wave equation is given below.

Maxwell’s equation is given as

∇²Ψ (x, y, z, t) − (1/ c²) (∂²Ψ (x, y, z, t)/ ∂t²) = 0

The Laplacian operator

∇2 = (∂²/ ∂x²) + (∂²/ ∂y²) + (∂²/ ∂z²)

The form of a wave might be represented by a function such as

Ψ(x) = f (2πx/ λ)

Euler's formula

e^±iθ = cos θ ± i sin θ

The analogous structure of the time-dependence of the wave

T(t) = f (2πνt)

A wave function of the form

Ψ (x, t) = exp [2πi ((x/ λ) − νt)]

Using the Planck formula (E = hν) and de Broglie formulas (p=h/ λ)

Ψ (x, t) = exp [i (px−Et)/ ℏ]

Where 

ℏ ≡ h/ 2π

The time derivative version becomes

∂Ψ/ ∂t = − (iE/ ℏ) exp[i(px−Et)/ ℏ]

iℏ (∂Ψ/ ∂t) = EΨ

−iℏ (∂Ψ/ ∂x) = pΨ

−ℏ2 (∂²Ψ/ ∂x²) = p²Ψ

The energy and momentum for a non-relativistic free particle

E = mv²/ 2 = p²/ 2m

iℏ (∂Ψ/ ∂t) = −(ℏ²/ 2m) (∂²Ψ/ ∂x²)

The energy and momentum for a nonrelativistic free particle

E = (p²/ 2m) + V(x)

The time dependent Schrodinger equation in 1D

iℏ (∂Ψ (x, t)/ ∂t) = [−(ℏ²/ 2m) (∂²/ ∂x²) + V(x)] Ψ (x, t)

The time dependent Schrodinger equation in 3D

iℏ (∂/ ∂t) Ψ (r⃗, t) = [−(ℏ²/ 2m) ∇² + V (r⃗)] Ψ (r⃗,t)

Time-Independent Schrodinger's Wave Equation

The time-independent form of Schrodinger’s wave equation describes the characteristics of stationary states, which are wave functions that have formed standing waves. The expression for the time-independent form of Schrodinger’s wave equation is given as follows.

Ĥ | Ψ) = E | Ψ)

where

• E is the energy of the system
• Ĥ is an observable, the Hamiltonian operator

The time-independent form of Schrodinger’s wave equation is only applicable if the Hamiltonian itself is not dependent on time explicitly. However, even then, the overall wave function is still dependent on time. The derivation for the time-independent form of Schrodinger’s wave equation is given below.

The time-dependent Schrodinger equation in 3D is given as

iℏ (∂/ ∂t) Ψ (r⃗, t) = [−(ℏ²/ 2m) ∇² + V (r⃗)] Ψ (r⃗, t)

For conservative systems, the energy is a constant, and the time-dependent factor form can be separated from the space-only factor

Ψ (r⃗, t) = ѷ (r⃗)e^−iEt/ℏ

The time independent Schrodinger equation is given as 

[− (ℏ²/ 2m) ∇² + V(r⃗)] ѷ(r⃗) = E ѷ (r⃗)