Time-Independent Schrödinger Wave Equation

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. 

(δ²/δx²)u = 1/v².(δ²/δt²)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;

(δ²/δx²) u = (δ²/δx²) 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;

(δ²/δt²) u = -ψ_x ω² (cos (ω.t))  →  (iii)

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

(δ²/δx²) ψ . cos (ω.t) = 1/v². (-ψ_x ω² cos (ω.t))

[(δ²/δx²) ψ] + [1/v². ψ_x ω²] = 0

This equation is called the wave equation.

As we know that ω = 2πf and v = f.λ

Putting in the above equation,

((δ²/δx²) ψ) + (4π² / λ²) ψ_x = 0  →  (iv)

In order to find out the value of λ², the sum of kinetic and potential energy is required.

E_total = K.E + P.E

• K.E = p²/2m
• P.E = V

E_total = p²/2m + V

Interchanging the equation,

E_total – V = p²/2m 

p² = (E_T – V) 2m

According to de-Broglie wave equation,

λ = h / p

λ² = h² / p²

Putting value of p², we get;

λ² = h² / (E_T – V)2m

1 / λ² = (E_T – V)2m / h²

Putting value of 1 / λ² in equation (iv), we get;

(δ²/δx²) ψ + (4π².(E_T – V).2m/h²) ψ_x = 0

(δ²/δx²) ψ + (8mπ².(E_T – V) / h²) 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.

(δ²/δx²) ψ + (8mπ² / h²) E_T – (8mπ² / h²) V ψ_x = 0

(δ²/δx²) ψ (h²/8mπ²) – V ψ_x = E . ψ_x

[(δ²/δx²) (h²/8mπ²) – V] ψ_x = E . ψ_x

As, (δ²/δx²) (h²/8mπ²) – 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 (δ²/δx²) + (δ²/δy²) + (δ²/δz²) 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.

References

• Physical Chemistry | Second edition by Keith J. Laidler (University of Ottawa) and John H. Meiser (Ball State University)
• Time Independent Schrodinger Equation (hyperphysics)
• The Time Independent Schrödinger Equation (quantummechanics.ucsd.edu)