Question

derive wave equation for class 9​

Answer 1

Answer:

Explanation: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 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.