Derivation Of Schrödinger Wave Equation

Schrödinger 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 Schrödinger Equation has two forms the time-dependent Schrödinger Equation and the time-independent Schrödinger Equation. The time-dependent Schrödinger Wave Equation derivation is provided here so that students can learn the concept more effectively.

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

Derivation Of Schrödinger Wave Equation

Schrödinger Wave Equation Derivation (Time-Dependent)

Considering a complex plane wave:

Derivation Of Schrödinger 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 a 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 is the derivation of Schrödinger 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.

6 Comments

  1. It is self explanatory

  2. zedekia Ojwaya Otieno

    I like it…..its justb very owesome

  3. This derivation is good and simple so easy to learn

  4. easy explanation

  5. Olusola Elijah Oluwatoyin

    This is awesome
    This is very educative

  6. This helped me in my endeavour to reactivate my long-dormant knowledge of quantum mechanics, thanks.

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