Question

explain schoridiger time independent wave equation for the partical in a box​

Answer 1

Explanation:

$\frac{d {}^{2} + \gamma }{ x } + \frac{d {}^{2} + \gamma }{y} + \frac{d {}^{2} + \gamma }{z} + (e - v) = 0 \\ thats \: a \: schoridiger \: wave \: equation... \\ hope \: its \: helps...$

Answer 2

Answer:

Hi !

Well, fundamentally for constructing and deriving the equation for the particle , we need to first know the energy permissible and also that the motion is allowed in how many dimensions?

For simplicity I'll consider the motion of the particle to be in 1 dimension along the x - axis.

And, considering the potential to be infinite at ends of the box at coordinates x = 0 and x = L

So, this implies that the motion is allowed in anywhere between x = 0 to x= L

Ok !

Now the comes the math,

We can simply consider the equation for the particle to be a standing wave, at some fixed time suppose at t=0 at some arbitrary position x where x€(0,L)

Now,

y = A sin (2πx/L)

y is the wave function here

So, that it remains as a standing wave.

A is the amplitude of the wave.

Now,

Differentiation of this with respect to time twice will give us

d2y/dx^2 = -4 π^2/L^2 * A sin(nπx/L)

which leads us to the form

a = -n^2π^2/L^2 * y

Let

2π/L = k

So,

this becomes

d^2y/dx^2 = -k^2 y

Now,

Schrodinger's time independent equation requires us to first identify and formally write down it's energy distributions ajd finally find out a satisfactory equation which can further lead us to a need of normalized wave function as it's solutions.

Alright !

Now,

We know,

From De-Broglie

mv = h/L

m^2v^2 = h^2/L^2

1/2mv^2 = h^2/2mL^2

So,

Now,

d^2y/dx^2 = -4π^2/L^2 * (y)

Now,

Total energy = KE + PE

KE = E - PE

1/2mv^2 = h^2/2mL^2 = (E-PE)

1/L^2 = 2m/h^2*(E-PE)

So,

d2y/dx^2 = -4π^2 (2m/h^2*(E-PE))*y

So,

this gives us the final time independent Schrodinger's equation

viz,

d^2y/dx^2 + 8π^2m/h^2 (E-PE)*y = 0

Now, this can be easily extended over the entire three dimensions giving us the general equation

d^2 (€)/dx^2 + d^2(€)/dy^2 + d^2(€)/dz^2 + 8π^2 m/h^2 (E-PE)*€ = 0

You can easily verify this by a little bit of math that the involvement of the second derivative of each of the two additional dimensions doesn't make a difference.

What only makes a difference here is the factors causing the partition of energy as well as the factors that are producing effective potentials in the system for interaction with the particle.

This complete Formalism we also call the Hamiltonian Formalism in classification and foundation of mathematically abstract but significant wavefunctions of particle in multiple dimensions.

I'm sorry to confuse you by introducing € this is what I've used to denote rhe wavefunction here instead where I initially used y but I forgot y is also used as an abbreivation for a dimension, so don't confuse the abbreviations y in both cases here.

Initially I used y for wavefunction of particle in one dimension whereas now it's €

Also, the most beautiful advantage of time independent Schrodinger's equation is that you can simply find out the normalized form of the wavefunction using the fact that total probability of finding the particle throughout the domain should be equal to summed up to 1 , in this case from x € (0,L).

Hope this helps you !