Applied Physics I: Chapter 7: Quantum Mechanics
Scroedinger time independent wave equation
It is convenient to use the time independent wave equation rather than using time dependent wave equation, because of the following reason.
SCHROEDINGER TIME
INDEPENDENT WAVE EQUATION
It
is convenient to use the time independent wave equation rather than using time
dependent wave equation, because of the following reason.
In
Schroedinger time dependent wave equation the wave function 'ᴪ' depends on
time, but in Schroedinger time independent wave function ᴪ does not depend on
time and hence it has many applications.
We
know that time dependent wave function
Splitting
the RHS of this equation into two parts, viz., (i) Time dependent factor and
(ii) Time independent factor, we get
where
ᴪ represents the time independent wave function (i.e.,) ᴪ = e‒ipx/h
Differentiating
equation (1) partially with respect to 't' we get
Differentiating
equation (1) partially with respect to 'x'
we get,
Differentiating
once again partially with respect to 'x'
we get
We
know the Schroedinger time dependent wave equation (one dimensional) is
We
can get the Schroedinger time dependent wave equation, just by substituting
equations (1), (2) and (3), which has relation between the time dependent wave
function (ᴪ) and time independent wave function (ᴪ), in equation (4).
∴ Substituting equations
(1), (2) and (3) in equation (4) we get
Equation
(5) represents the one dimensional (x‒direction)
Schroedinger time independent wave equation, because, in this equation the wave
funtion ᴪ is independent of time. Similarly, the three dimensional Schroedinger
time independent wave equation can be written as
Note:
For
a free particle, the potential energy V=0
∴ Schroedinger time
independent wave equation becomes
d2ᴪ/dx2 + (2m/ḧ2)Eᴪ
= 0. [for one dimension]
and
∇2ᴪ
+ (2m/ḧ2)Eᴪ = 0. [for 3 dimension].
Applied Physics I: Chapter 7: Quantum Mechanics : Tag: Applied Physics : - Scroedinger time independent wave equation
Applied Physics I: Chapter 7: Quantum Mechanics
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