Derivation of Schrodinger Wave Equation

The Schrödinger Equation is a mathematical equation that expresses how a physical quantity changes over time while accounting for quantum factors such as wave-particle duality. 

• The Schrodinger wave equation describes the behavior of a particle in a field of force or the change of a physical parameter over time.
• Erwin Schrödinger developed the equation and received the Nobel Prize in 1933.
• Schrodinger's wave equation is similar to Newton’s laws of motion but not as deterministic as Newton's law of motion.
• The possible positions of the subatomic particles observed can be predicted if the initial conditions are known.
• However, Newton's equations of motion may be used to determine the position of an item after a certain period if the beginning position of the object and the number of forces acting on it are known.
• The Schrödinger Equation exists in two forms: time-dependent and time-independent. 

Key Terms: Schrodinger's equation, Wave matter duality, Time-dependence, Schrodinger wave equation, Maxwell Equation, Position, Time, Electron, Atom

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What Is Schrodinger Wave Equation?

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The Schrodinger wave equation is a mathematical equation that describes the energy and position of an electron in space and time while accounting for the matter wave nature of an electron within an atom.

• The Schrodinger equation provides an in-depth explanation of the wave functions or probability waves that control the behavior of some smaller particles.
• The equation also shows how these waves are affected by external forces.
• Furthermore, the equation makes use of the energy conservation concept, which provides information on the behavior of an electron connected to the nucleus.
• Calculating the Schrödinger equation gives Ψ and Ψ₂, which help in determining quantum numbers, orientations, and shapes of orbitals, where electrons lie in molecules or atoms.

There are two Schrodinger wave equation

• Time-dependent Schrödinger equation
• Time-independent Schrödinger equation

The time-dependent Schrödinger equation is given by

\(i \hbar \frac{d}{d t}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle\)

Based on position, the time-dependent Schrödinger equation is given by

\(i \hbar \frac{\partial \Psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \Psi}{\partial x^{2}}+V(x) \Psi(x, t) \equiv \tilde{H} \Psi(x, t)\)

Where

• i = Imaginary unit
• Ψ = Time-dependent wavefunction
• h² is h-bar
• V(x) = potential and
• \(\hat{H}\) is Hamiltonian Operator

The time-independent Schrödinger equation in compressed form can be expressed as

\(\hat{H}\Psi = E \Psi\)

The time-independent-Schrödinger-nonrelativistic-equation is given by

\(\left[\frac{-\hbar^{2}}{2 m} \nabla^{2}+V(\mathbf{r})\right] \Psi(\mathbf{r})=E \Psi(\mathbf{r})\)

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Derivation of Schrodinger's Wave Equation (Time-Dependent)

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Let us consider a complex plane wave

\(\Psi ( x, t ) = Ae^{i(kx - \omega t)}\)

The Hamiltonian system is given by

\(H=T+V\)

Where

• V is the potential energy
• T is the kinetic energy
• H is the total energy

We can write the above equation as

\(E = \frac{ p^{2} }{ 2m } + V(x)\)

Now taking the derivates, we get

\(\frac{\partial \Psi }{ \partial t} = - i \omega A e^{ i(kx - \omega t)} =- i \omega \Psi (x, t)\)

Also

\(\frac{ \partial ^ { 2 } \Psi }{ \partial x^{2} } = -k^{2} Ae^{i (kx - \omega t)} = -k^{ 2 } \Psi (x, t)\)

Now, we know that

\(p = \frac{ 2 \pi \bar{h} }{ \lambda } \:and \: k = \frac{ 2 \pi }{ \lambda }\)

Where

• λ is the wavelength
• k is the wavenumber

We have, \( k = \frac{ p }{ \bar{h} }\)

Therefore, we can write

\(\frac{ \partial ^ { 2 } \Psi }{ \partial x^{2} } = - \frac{ p^{2} }{ \bar{h} ^ {2} } \Psi (x, t)\)

Now multiplying Ψ (x, t) by the Hamiltonian we get,

\(E \Psi (x, t) = \frac{ p^{2} }{2 m} \Psi (x,t) + V(x) \Psi (x, t)\)

The above equation can also be written as

\(E \Psi(x, t) = \frac{ - \bar{h} ^ {2} }{2m} \frac{\partial ^ { 2 }\Psi}{\partial x^{2}} + V(x) \Psi (x, t)\)

The equation of energy wave of a matter wave is written as

\(E = \bar{h} \omega\)

Therefore, we can write

\(E \Psi (x, t) = \frac{ \bar{h} \omega }{ - i \omega } \Psi (x, t)\)

Now, we can get the Schrodinger Wave Equation by combining the right parts

\(i \bar{h} \frac{\partial \Psi}{\partial t} = \frac{- \bar{h} ^{ 2 } }{2 m} \frac{\partial ^ {2} \Psi}{\partial x^{2}} + V(x) \Psi (x, t)\)

This is the derivation of the Schrödinger Wave Equation (time-dependent).

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Things to Remember

• The Schrodinger wave equation describes the behavior of a particle in a field of force or the change of a physical parameter over time.
• Schrödinger Equation exists in two forms: time-dependent and time-independent.
• The Schrodinger wave equation is a mathematical equation that describes the energy and position of an electron in space and time while accounting for the matter wave nature of an electron within an atom.
• Newton's rules are deterministic because they can predict how the forces will interact and, as a result, where the object will be at a later point in time.
• The Schrodinger equation is a differential equation that uses all of the spatial coordinates and time to explain the system in question.