Needs and justification of Schrodinger’s equation have an wide application in field of quantum mechanics. In this article we are going to discuss about Postulates of quantum mechanics and Needs and justification of Schrodinger’s equation along with some basic information about wave function and formulas.
Wave function- wave function is represented by ψ, it represents the amplitude of matter waves. It is complex quantity and represent the state of function , ψ (r,t).
Postulates of quantum mechanics:-
(1) ψ(r,t) Can completely describe a physical entity .
A wave function must satisfied the all conditions-
(a) ψ and dψ/dx must be finite for all the values of x.
(b) ψ and dψ/dx must be continuous for all values of x.
(c) ψ and dψ/dx must be single valued.
(2) About any point r, and time t the probability of finding the particle in a volume element dτ = dxdydz
P( r) dτ = |ψ(x,y,z)|²dτ
P( r) dτ =ψ (x,y,z,t) ψ* (x,y,z,t) dτ
If a particle is somewhere in space then its probability is one .
( Note – if particle is not present in space then its probability is zero)
∫p(r)dτ = ∫₋∞₊∞ ψ(x,y,z,t) ψ* (x,y,z,t) dτ
If ψ₁ and ψ₂ are two different wave function for a given system and the probability, ∫ψ₁ψ₂*dτ vanishes or zero , then this condition is called orthogonal condition.
(3) every observable quantity can be expressed by an operator in quantum mechanics . If Â si an operator for observable quantity and its Eigen value is ‘a’ can be find by acting an operator Â on the wave function ψ.
Âψ = aψ
Where a is Eigen value
ψ is Eigen function
And this is said to be Eigen equation.
(4) A particle which is moving in space can be written asoSchrodinger’s wave equation.
∇²ψ + 2m /
h² ( E-V) ψ= 0
(5) Expectation value/ Average value – expectation value for any physical quantity A,
<A> = ∫₋∞⁺∞ ψ* Â ψdτ / ∫₋∞⁺∞ ψ*ψdτ
Needs and justification of Schrodinger’s equation-
Schrodinger represent a equation to describe the quantum theory which is basic fundamental of classical mechanics. It must satisfied the following condition-
- It must be linear so that any combination of a given solution can also be a solution of the equation. the phenomena of interference, diffraction produced by particles, as also the possibility of formation of wave packets can be explained by this condition.
- The cofficients of the wave equation must involve only fundamental constants like Planck’s constant h ,mass m or charge q of a particle, and not the parameters (internal) concerning a particular kind of motion of the Particle ( like energy and momentum) or angular frequency ω and propagation constant or wave number k. Because the presence of these parameters even in a superimposed solution cannot be a solution of the equation. Since the differential equations are easiest to handle , it is worthwhile to try it first and it may turn out that the requirements can be met by a differential equation.
The wave of a particle can be explained If particle is moving along positive x-direction with momentum p and energy E ,
ψ(x,t) = cos(kx-ωt) or ψ(x,t) =sin (kx-ωt) (1)
ψ(x,t) =exp i (kx-ωt) or ψ(x,t) =exp -i (kx-ωt) (2)
Or some suitable combination of them.
The wave function defined by any one given by above equation satisfied a number of wave equations and we try one as given below
δ²ψ/δt² = γδ²ψ/ δx² (3)
The de-Broglie relations are –
P = h/ λ and E= hν
Let us denote p and E in terms of
h = h/2π
Then p =
hk where k= 2π/λ
K being the wave number or propagation constant and
hω , where ω= 2πν
Substituting the form of ψ given by earlier equation or their linear combination would satisfy equation (3) provided
γ = ω²/k² = E²/p² = p²/4m² (4)
Assuming dp/dt =0 i.e. absence of force,m is the mass of the particle .the requirements (4) involves parameters of motion , energy E, and momentum p. We would, however, prefer a wave equation that depends only on basic properties like mass m, or charge q etc. equation (3) can be discarded. Another equation then could be
δψ/δt = γ δ²ψ/δx² (5)
(1)type of solution cannot satisfy the above equation . The wave function of the exponential type given in (2) will satisfy this equation provided
γ = i
h /2m (6)
It may be noted that the exponential form for the wave function is a representation of a plane monochromatic wave . The quantum mechanical wave equation for one dimension with γ given by equation (6) would be
δψ/δt = i
h δ²ψ/ 2mδx²