Introduction

In quantum mechanics, Schrodinger equation and its solution is a mathematical equation which describes the changes over time of a physical system. This equation was given by Erwin Schrodinger in 1925 and published it in 1926. He was also awarded by Nobel Prize in 1933. In this article, we are going to discuss the Schrodinger equation and its solution widely along with Their mathematical formulas.

Schrodinger equation and its solution

Schrodinger equation – time dependent

Let us assume a wave moving in x-direction , having momentum P and energy E,

ψ(r,t ) = Aexpi (kx-ωt)        (1)

We know,

k = 2π/λ

ω = 2πν

E = hν

And P = h/ λ

P = Aexp i ( 2πx/λ − 2πνt)

∴ ω   = 2πE / h

k = 2π/λ

ψ(x,t) = A exp i (px- Et )/ h          (2)

Now differentiate this equation with respect to time t,

dψ/dt = A exp i (px-Et) /h (-E)i/h 

dψ/dt = –iE ψ/ h

Eψ = h δψ/δt          (3)

Differentiate equation (2) two times with respect to x,

dψ/dx =  A exp i /h (px-Et) . iP / h

dψ/dx =i/ h Ap exp i / ( px-Et )

d²ψ/dx² = (i ²A/ h²) P² exp i( px-Et) / h

P²ψ  = − h² d²ψ/ dx²     (4)

Total energy of particle

E= P²/ 2m + V

Now multiply by ψ by RHS,

Eψ =P ²ψ/2m + Vψ

Now put the values from (3) and (4),

i h dψ/ dt = – (h²/2m ) d²ψ/ dx²+  Vψ

This is Schrodinger equation for time dependent.

Hψ = Eψ

For free particle,

V = 0

Then above equation can be written as-

ih dψ/dt = –h²d²ψ  / 2m dx²       (5)

Above equation is Called Schrodinger equation for free particle.

Schrodinger equation for time independent

In many situations the potential energy of a particle does not depend on time but depends only on the position of particle. The force acting on particle depends only on its position. The wave function ψ(r,t ) may be written as a product of a function φ(t) that depends only on time and u(r) that depends only on r.

i.e.       ψ(r,t ) =φ(t)u(r)       (1)

Let us consider a equation

ψ( x,t ) = Aexp i (Px -Et) / h

Again,  ψ( x,t ) = Aexp i Px / h . exp (-Et/ h)

Now,  ψ( x,t ) = ψ(x) exp- i Et / h       (2)

Differentiate this equation with respect to time t,

dψ (x,t) /δt = ψ(x)  exp (-iEt / h) {-iE/ h  }    (3)

Now differentiate equation (2) with respect to x, two times

d²ψ(x,t)/ dx² = d²ψ(x)/ dx² ( exp( –iEt/ h dφ/dt) )    (4)

We know that time dependent Schrodinger’s equation is-

h dψ/dt = – h² / 2m (d²ψ(x)/ dx²) + Vψ

Now put the values from (3) and (4) ,

hψ(x) exp (- iEt / h) = –h²/2m

(d²ψ(x,t)/ dx²  ( exp( -iEt/ h) ) +Vψ )

d²ψ(x)/dx² + 2m/ h² (E-V) ψ(x) =  0      (5)

This is call Schrodinger equation for time independent.

For free particles  v=0

Hence the above equation becomes

d²ψ(x)/dx² + 2m/ h² (E) ψ(x) = 0

Must read

particle in a three dimensional box

Particle in a three dimensional potential box

Solution of schrodinger equation potential step

Solution of Schrodinger equation for a potential step

particle in one dimensional box

Particle in one dimensional potential box

Application of schrodinger equation 

Application of Schrodinger equation

Time-independent Schrodinger equation and stationary state solution

If in a particular state the probability distribution function doesn’t depend on time then the state of function is called stationery state.

Time dependent Schrodinger’s equation-

h dψ/dt=  – h²/2m d²ψ/ dt²+ Vψ

– h²/2m (d²ψ/dr (r,t) )+ V(r) ψ(r,t) = i hdψ(r,t)/dt      (1)

Where ψ(r,t)  = ψ(r)φ(t)          (2)

Put this in equation (1)

[- h²/2m( ∇²)+ V(r) ] ψ(r)φ(t)   = ihd/dt {ψ(r)φ(t) }

Or,

φ(t)[-h²/2m(∇²ψ(r) )+V (r)ψ(r) ] = ψ(r) { ihdφ/dt}

Now devide by,

φ(t)/ψ(r)φ(t) [ –h²/2m( ∇²ψ(r))+V(r)ψ(r)] = ψ(r)/ψ(r)φ(t) {idφ/dt}

Let , E = 1/ψ(r) [- h²/2m (∇²ψ(r) + V(r) ψ(r)] = 1/φ(t) { ih dφ/dt}

Let us suppose that,

E = 1/ψ(r) [ – h²/2m (∇²ψ(r))+ V(r)ψ(r) ]

Also, E =1/φ(t) { ih dφ/dt}

i hdφ/dt = Eφ(t)

dφ(t)/φ (t) = Edt

∫dφ(t)/φ(t) = ∫Edt / ih

log φ(t) = A ( -Et/ ih)

φ(t) = Aexp (-iEt/h)

ψ(r,t) = ψ(r)φ(t)

Also, ψ(r,t) =A ψ(r)exp(-iEt/h)

Now,

ψ(r,t) = Aψ(r)exp(-iEt/ h)

For convincing include A into ψ(r)

ψ(r,t) = ψ(r)exp (-i/hEt)

Probability density of Schrodinger equation and its solution

P(r,t) = |ψ(r,t)|²

P(r,t) = ψ(r,t)ψ*(r,t)

Put the value from above equation

ψ*(r,t) = ψ(r) exp iEt/ h

Hence,

P(r,t) =ψ(r)exp (-iEt / h)ψ(r,

Conclusion

In this particular article ,Schrodinger equation and its solution we have discussed about the  equation Schrodinger for time dependent and time independent along with its derivations in most easiest way possible.

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