In 1926, Austrian physicist Erwin Rudolph Joseph Alexander Schrödinger presented a mathematical concept of wave-particle dualism. He thought that the electrons have standing waves just like waves of a stretched string. This was a new idea of that time that gave rise to the development of wave mechanics to tackle the problems in both atomic and molecular physics and chemistry.

Schrodinger described how the quantum state of a physical system changes with time in terms of partial differential equations. This equation is known as the Schrodinger wave equation. In quantum mechanics, as every Newtonian variable has an alternative, the analog of Newton’s law of motion is Schrodinger’s wave energy equation.

This equation is not a simple algebraic equation, but in general, a linear partial differential equation, describing the time-evolution of a system’s wave function. The Schrodinger wave equation is an equation, which expresses energy in the form of the wave function (ψ) in different physical conditions.

The wave-like behavior of microscopic particles is far beyond the reach of our senses. But, credit goes to Schrödinger who developed an equation that clarifies all our confusions. The main concepts given by this equation are given below:

- The motion of electrons describes the atomic model.
- The harmonic oscillators define an atom’s energy levels.
- The diatomic molecule can be defined by rotational and vibrational energy levels.

There are two types of Schrodinger wave equations:

- Time-dependent Schrodinger wave equation
- Time independent Schrodinger wave equation

The derivation of Schrodinger’s wave equation (time-independent) is given below.

## Derivation of Schrödinger Wave Equation

A uniform wave is produced in a vibrating string, fixed at both ends. Now as Schrodinger proposed that electrons have standing waves, we have to start with sin functions of standing waves.

Let us take a linear partial differential equation.

(δ^{2}/δx^{2})u = 1/v^{2}.(δ^{2}/δt^{2})u → (main equation)

where,

v is the velocity with which disturbance moves along a string. The boundary conditions for this supposition are taken as x = 0 and x = 1.

Now;

u (x,t) = ψ_{x }cos (θ)

where θ = t

This equation has two variable entities, x, and t. Therefore, we will differentiate the variables one by one.

u (x,t) = ψ_{x }cos (t) → (i)

Differentiating the above equation (i) with respect to variable ‘x’, we get;

(δ/δx) u = (δ/δx) ψ . cos (ω.t)

Differentiating once again, we get;

(δ^{2}/δx^{2}) u = (δ^{2}/δx^{2}) ψ . cos (ω.t) → (ii)

Now differentiate eq (i) with respect to ‘t’, we get;

As, d/dx cos (x) = -sin (x)

(δ/δt) u = -ψ_{x }(ω.sin(ω.t))

Differentiating once again, we get;

(δ^{2}/δt^{2}) u = -ψ_{x }ω^{2} (cos (ω.t)) → (iii)

Putting equations (ii) and (iii) back in equation (i), we get;

(δ^{2}/δx^{2}) ψ . cos (ω.t) = 1/v^{2}. (-ψ_{x }ω^{2 }cos (ω.t))

[(δ^{2}/δx^{2}) ψ] + [1/v^{2}. ψ_{x }ω^{2}] = 0

This equation is called the wave equation.

As we know that ω = 2πf and v = f.λ

Putting in the above equation,

((δ^{2}/δx^{2}) ψ) + (4π^{2} / λ^{2}) ψ_{x} = 0 → (iv)

In order to find out the value of λ^{2}, the sum of kinetic and potential energy is required.

E_{total }= K.E + P.E

- K.E = p
^{2}/2m - P.E = V

E_{total }= p^{2}/2m^{ }+ V

Interchanging the equation,

E_{total }– V = p^{2}/2m^{ }

p^{2} = (E_{T }– V) 2m

According to de-Broglie wave equation,

λ = h / p

λ^{2 }= h^{2 }/ p^{2}

Putting value of p^{2}, we get;

λ^{2 }= h^{2 }/ (E_{T }– V)2m

1 / λ^{2 }= (E_{T }– V)2m / h^{2}

Putting value of 1 / λ^{2 }in equation (iv), we get;

(δ^{2}/δx^{2}) ψ + (4π^{2}.(E_{T }– V).2m/h^{2}) ψ_{x} = 0

(δ^{2}/δx^{2}) ψ + (8mπ^{2}.(E_{T }– V) / h^{2}) ψ_{x} = 0

The above equation is the time-independent Schrodinger wave equation along the x-axis in a one-dimensional box. This equation can be simplified into different forms.

By separating E and V from the above equation, we get the famous form of the Schrodinger wave equation.

(δ^{2}/δx^{2}) ψ + (8mπ^{2} / h^{2}) E_{T }– (8mπ^{2} / h^{2}) V ψ_{x} = 0

(δ^{2}/δx^{2}) ψ (h^{2}/8mπ^{2}) – V ψ_{x }= E . ψ_{x}

[(δ^{2}/δx^{2}) (h^{2}/8mπ^{2}) – V] ψ_{x }= E . ψ_{x}

As, (δ^{2}/δx^{2}) (h^{2}/8mπ^{2}) – V = Ĥ, this equation becomes;

Ĥ . ψ_{x} = E . ψ_{x}

where,

- Ĥ = Hamiltonian operator
- E = Energy of wave
- ψ
_{x}= Wave function of particular wave in one dimension (x)

### For 3-D system

For the conversion of the one-dimensional time-independent Schrodinger wave equation into two or three dimensions, the differential (δ/δx) is enhanced by the addition of (δ/δy) and (δ/δz) dimensional operators. The (δ^{2}/δx^{2}) + (δ^{2}/δy^{2}) + (δ^{2}/δz^{2}) operators sum up to form the ‘Laplacian operator (∇^{2})’, an important quantum mechanical operator.

The final equation for 3-D becomes the one without any reference to any specific axis.

Ĥ . ψ = E . ψ

## Applications of Schrodinger wave equation

- Schrodinger’s wave equation well explained the structures of atoms and molecules.
- It describes the shapes of orbitals and their orientation.
- The solution of the equation gives information about the quantization of energy.
- Most of all, this equation is used to figure out the probability of finding particles in 1-D, 2-D, and 3-D boxes or wells which gives rise to several other applications.
- Schrodinger wave equation is used to figure out the allowed values for a quantum mechanical system.
- The probability of finding a particle at a certain position is given by this equation.
- Particles in a box are one of the best applications of Schrodinger wave equations.
- This equation becomes the basis of all types of wave mechanics.
- Scanning Tunneling Microscope (STM) is an application of the Schrodinger wave equation.
- Quantum tunneling is an advanced featured application of this equation, etc.

## Key Takeaway(s)

## Concepts Berg

**What is the Schrodinger wave equation?**

Schrodinger wave equation is a mathematical equation that describes the energy and position of electrons in space and time when the matter-wave nature of electrons is taken into account. It is a linear partial differential equation that determines the wave function of the quantum mechanical system.

**What is the largest number of independent variables used as part of a partial differential equation used to represent a physical phenomenon?**

The partial differential equation is a mathematical equation that contains a derivative of one variable with respect to the derivative of two or more independent variables. So the largest number of independent variables in the partial differential equation is two but it can be greater than two.

**How do I show that a wave function satisfies the time-independent Schrodinger equation?**

The wave function whose all observable properties are constant must satisfy the time-independent Schrodinger wave equation.

**What is the physical significance of the time-dependent and independent Schrodinger wave equation explained deeply?**

Schrodinger wave equation whether it is time-dependent or independent helps us:

- To find the wave function
- The solution of the equation gives the probability of a particle in a box

**How would one know if a wave function satisfies the time-independent Schrodinger equation?**

The wave function whose all observable properties are constant must satisfy the time-independent Schrodinger wave equation. If the wave function is an eigenfunction of the Hamiltonian operator, it satisfies the Schrodinger wave equation.

**What is the conclusion for Schrodinger’s time-independent equation?**

The solution of the Schrodinger wave equation is used to find the problems in general. It tells us if a particle is trapped in a finite well and if its energy has some discrete values.

**What is Schrodinger’s 3-D equation?**

Schrodinger wave equation in three dimensions is given as;

E ψ = Ĥ ψ

**What does the Schrodinger equation represent?**

The Schrodinger wave equation represents the total energy of a system. It gives the probability of finding a particle in a deep potential well.

**What are the eigenfunctions in a Schrodinger wave equation?**

If an operator is applied to some function, and we get the same function back, the function is called an eigenfunction with some constant value. These constant values are called eigenvalues. In the Schrodinger wave equation, Ĥ ψ = E ψ (Ψ) is an eigenfunction.

**What is psi in Schrodinger’s wave equation?**

Psi is a wave function representing the spatial distribution of particles. It completely defines the system. If psi is known, we can determine any observable property of the system.

**Reference Books**

**Physical Chemistry**| Second edition by**Keith J. Laidler**(University of Ottawa) and**John H. Meiser**(Ball State University)

**Reference links**

- Time Independent Schrodinger Equation (hyperphysics)
- The Time Independent Schrödinger Equation (quantummechanics.ucsd.edu)