Question

Derive Schrodinger equation

Answer 1

Answer:

Explanation:

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.

Answer 2

Let's consider the wave of the electron considering a point in space where our analysis starts at t=0
So,.
The wave can be written as,

Wavefunction = A sin(2πx/f)
where f is the wavelength

So,
d(Wavefunction)/dx = 2πA/f cos(2πx/f)
Again differentiating,

d2(Wavefunction)/dx^2 = -4π^2A/f^2 sin (2πx/f)
Therefore,
Let Wavefunction be represented by y
So,
d^2 y / dx^2 = - 4π^2/f^2 * y
Now,
By De-Broglie
f = h/mv
So,
f^2 = h^2 / m^2 v^2
Now,
mv^2 = h^2/mf^2
So,
Kinetic Energy = h^2 / 2mf^2

Now,
Total energy = Potential energy + Kinetic energy
So,
1/2mv^2 = Total energy - Potential energy

But,
f^2 = h^2/2m(Kinetic energy)

So,
Putting this value in original equation
d^2y/dx^2 = -8π^2m(Kinetic energy)/ h^2
So,
d^2y/dx^2 = -8π^2 m(E-PE) / h^2
Now,
Let us again represent Wavefunction not as y

So,
(Wavefunction)"(x) = -8π^2m(E-PE)/h^2
Now, from the Quantum Mechanical assumption we know that the probability is equal for all directions, here not assuming more than three dimensions of space, we can write

(Wavefunction)"(x) + (Wavefunction)"(y) + (Wavefunction)"(z) = -8π^2*m(E-PE)/h^2

This is the Time independent Schrodinger equation.
However, The Schrodinger equation can be computed in almost I'm possible number of ways !!
This is one such form of the Equation.

Hope this helps you !