Derivation Of Schrodinger Wave Equation

Derivation Of Schrodinger Wave Equation

Schrodinger Equation is a mathematical expression which describes the change of a physical quantity over time in which the quantum effects like wave-particle duality are significant. The Schrodinger Equation has two forms the time-dependent Schrodinger Equation and the time-independent Schrodinger Equation. The time-dependent Schrodinger Wave Equation derivation is provided here so that students can learn the concept more effectively.

Questions related to the derivation of Schrodinger Wave Equation is one of the most commonly asked questions in board exams and various competitive exams. The derivation of Schrodinger Wave Equation is given below in such a way that students understand the concept in an interesting and easy manner.

Schrodinger Wave Equation Derivation (Time-Dependent)

Considering a complex plane wave:

Derivation Of Schrodinger Wave Equation

Now the Hamiltonian of a system is

Derivation Of Schrodinger Wave Equation

Where ‘V’ is the potential energy and ‘T’ is the kinetic energy. As we already know that ‘H’ is the total energy, we can rewrite the equation as:

Derivation Of Schrodinger Wave Equation

Now taking the derivatives,

Derivation Of Schrodinger Wave Equation

We know that,

Derivation Of Schrodinger Wave Equation

where ‘λ’ is the wavelength and ‘k’ is the wavenumber.

We have

Derivation Of Schrodinger Wave Equation

Therefore,

Derivation Of Schrodinger Wave Equation

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

Derivation Of Schrodinger Wave Equation

The above expression can be written as:

Derivation Of Schrodinger Wave Equation

We already know that the energy wave of matter wave is written as

Derivation Of Schrodinger Wave Equation

So we can say that

Derivation Of Schrodinger Wave Equation

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

Derivation Of Schrodinger Wave Equation

This was the Derivation Of Schrodinger Wave Equation (time-dependent). Students must learn all the steps of Schrodinger Wave Equation derivation to score good marks in their examination. Stay tuned with BYJU’S and learn various other derivation of physics formulas.


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