Schrodinger Wave Equation and Derivation

In this chapter, you will learn about the Schrodinger wave equation and its derivation. A microscopic object, such as an electron exhibits both observable wavelike and particle-like characteristics. Classical mechanics, which are based on Newton’s laws of motion, cannot explain the behavior of such particles.

In order to explain the behavior of electrons and other microscopic particles, a new branch of science known as quantum mechanics or wave mechanics had developed. Quantum mechanics is a theoretical science which deals with the dual nature of matter.

Based on the quantum mechanics, an Austrian physicist, Erwin Schrodinger, developed a new atomic model in 1926. This model is known as wave mechanical model of the atom or quantum mechanical model of atom, which is based on the particle and wave nature of the electron.

In this model, the behavior of the electron in an atom is explained by a differential equation, which is known as Schrodinger wave equation. This equation can be expressed in several forms one of which is given below:

\[ \frac{d^2 \psi}{dx^2} + \frac{d^2 \psi}{dy^2} + \frac{d^2 \psi}{dz^2} + \frac{8 \pi^2 m}{h^2} (E – V) \psi = 0 \]

In the above Schrodinger wave equation,

x, y and z are cartesian coordinates of the electron,
m is the mass of the electron,
E is the total energy of the electron,
V is the potential energy of the electron,
h is Planck’s constant,
ψ (Greek letter psi, pronounced as si) is a wave function which represents the amplitude of the electron wave at various points surrounding the nucleus. This amplitude is expressed in terms of x, y, z space co-ordinates of the electron with respect to the nucleus at 0, 0, 0.
d²ψ/dx² refers to the second derivative of ψ with respect to x only and so on.

Derivation of Schrodinger Wave Equation

Erwin Schrodinger, in his atomic model, described the electron as a three-dimensional wave in the electric field of a positively charged nucleus. He derived an equation which describes the wave motion of an electron propagating in three-dimensional space. This wave equation is called Schrodinger’s wave equation.

The main problem Schrödinger aimed to solve was the calculation of the probability of finding an electron at various points within an atom. His equation is based on the concept of the electron behaving as a “standing wave” around the nucleus. The equation for the standing wave is expressed as:

ψ = A sin2πx/λ – – – – (a)

Here,

ψ (pronounced as sigh) is a mathematical function representing the amplitude of wave known as wave function,
x is the displacement in a given direction,
λ is the wavelength,
A is a constant.

By differentiating equation (a) twice with respect to x, we get

dψ/dx = (A2π/λ) cos2πx/λ – – – – (1)

and d²ψ/dx² = – (A4π²/λ²) sin2πx/λ – – – – (2)

But A sin2πx/λ = ψ

∴ d²ψ/dx² = – (4π²/λ²)ψ – – – – (3)

The kinetic energy of the particle of mass m and velocity ν is given by the relation:

K.E. = mv²/2 = m²v²/2m – – – – (4)

According to Broglie’s equation, λ = h/mv

or λ² = h²/m²v²

or m²v² = h²/λ²

Substituting the value of m²v², we have

K.E. = h²/2mλ² – – – – (5)

From equation (3), we have

λ² = – (4π²ψ/d²ψ/dx²) – – – – (6)

Substituting the value of λ² in equation (5), we get

K.E. = -(h²d²ψ)/(2m*4π²ψ*dx²)

K.E. = -(h²d²ψ)/(8π²mψdx²)

The total energy E of a particle is the sum of kinetic energy and the potential energy. Thus,

E = K.E. + P.E.

or K.E. = E – P.E.

or -(h²d²ψ)/(8π²mψdx²) = E – P.E.

or d²ψ/dx² = -8π²mψ(E – P.E.)/h²

or d²ψ/dx² + 8π²mψ(E – P.E.)/h² = 0

This is Schrödinger’s equation in one dimension. It needs to be generalized for a particle whose motion is described by three spatial coordinates, such as x, y, and z. Thus, the equation becomes:

\[ \frac{d^2 \psi}{dx^2} + \frac{d^2 \psi}{dy^2} + \frac{d^2 \psi}{dz^2} + \frac{8 \pi^2 m}{h^2} (E – P.E.) \psi = 0 \]

This equation is the time-independent Schrödinger wave equation in three dimensions. The first three terms on the left-hand side in the above wave equation are represented by Δ²ψ (pronounced as del-square sigh).

\[\nabla^2 \psi + \frac{8 \pi^2 m}{h^2}(E – P.E.) \psi = 0\]

Where,

Δ² = d²/dx² + d²/dy² + d²/dz² represents the Laplacian operator,
ψ is the wave function,
m is the mass of the particle,
E is the total energy,
P.E. is the potential energy,
h is Planck’s constant.

Sometimes the Schrodinger wave equation is written in the form Hψ = Eψ. Here, H is a mathematical operator known as Hamiltonian operator.

The Schrodinger wave equation given above is a second-order differential equation and may be solved for ψ by using standard methods. It has several solutions when the Schrodinger wave equation is solved precisely for hydrogen atom and helium ion (He+). Only certain solutions from these are valid because some of these are imaginary. The valid solutions thus obtained explain the behavior of the electron in H atom and He+ ion.

The solutions of the wave equation are known as eigen functions or wave function which is always finite, single valued and continuous. It is zero at an infinite distance. If the potential energy term in the wave equation is known, the total energy E and the corresponding wave function ψ can be determined.

Significance of ψ and ψ²

We know that the moving electron exhibits wave-like behavior and a wave is completely defined by its amplitude. Therefore, ψ represents the amplitude of the electron wave. However, ψ itself has no direct physical significance.

The square of ψ i.e. ψ² has a physical significance. Just as light radiations where the square of amplitude of the wave gives the intensity of light, similarly, in electron wave, ψ² gives the intensity of electron at any point.

In other words, the square of ψ i.e. ψ² describes the maximum probability of finding an electron in a particular region. That is, it gives probability density. The space in which there is maximum probability of finding an electron is termed as orbital. Thus, ψ² is called probability density, while ψ is referred to as probability amplitude.

Thus, the solutions of Schrodinger wave equation have replaced the discrete energy levels or orbits proposed by Niels Bohr and introduced the concept of the most probable regions in space in terms of ψ2. A large of ψ2 indicates a high probability of finding the electron at that location, while a small value of ψ2 means low probability.

If ψ2 is nearly zero at a certain point, it implies that the probability of finding electrons at that point is negligible. Therefore, the wave mechanics approach provides the meaningful wave functions that describe both position and energy levels of electrons in an atom.