Derivations
Understanding Schrödinger’s Time-Dependent Equation and need of it!!!
In my previous article about
Schrödinger’s equation, I thoroughly derive Schrödinger’s Time-Independent equation, but still now we are not independent to understand our quantum world
fully!!!
For that we need another and more
sensible version of Schrödinger’s wave equation. Any sensible wave equation
should be both space and time dependent. In the preceding derivation, time
dependence was overlooked by concerning ourselves only with derivatives of y
with respect to x. In doing so, any knowledge of the direction sense of the
wave pattern was forgone.
But there is no harm to derive and
learn the previous derivation as it will behave like a pseudo-derivation of
time-dependent SE.
Now, to derive time-dependent SE, we
need knowledge of some equations which are as follows:
λ=h/p
(de-Broglie’s Wavelength)
E=hv (Plank’s
Energy-Frequency Relation)
w=2πv (Definition
of Angular Frequency)
Where, λ= wavelength
h=plank’s Constant
p=momentum
E=Energy
v=frequency
Now, as we deed in last derivation, we will firstly take the prototype
wave-function,
Ψ=Asin (px/ ħ-wt) …………….(1)
Using Plank’s Energy-Frequency Relation and Definition of Angular
Frequency,
Ψ=Asin [(px-Et)/ ħ] ………………(2)
Now, derivative of equation 2 with respect to time,
∂Ψ/∂t = (-E/ ħ ) A cos[(px-Et)/ ħ] …………….(3)
But equation 3 has a serious problem. Here, wave-function is represented
by a function. It should include wave-function only, not its function as it
should be independent of the form of any of its particular solution.
To rectify this, we will use one identity of basic trigonometry,
sin^2 θ +cos^2 θ=1
Now, A^2 sin^2[(px-Et)/ ħ] + A^2 cos^2
[(px-Et)/ ħ] =A^2 ………..(4)
From equation 2 and 4, we get,
A cos [(px-Et)/ ħ] =±√A^2- Ψ^2
……………..(5)
From equation 5
and 3,
E= [∓ ħ/√A^2- Ψ^2](∂Ψ/∂t)
Substituting this
value to the right side of the time independent equation,
- ħ^2/2m(∂^2
Ψ /∂x^2) + V(x)y=[∓ ħ/√A^2- Ψ^2](∂Ψ/∂t)
But this result
has many serious problems:
1) Amplitude of
the wave should not appear in what is presumed to be a physical
law
2) Presence of
square root in denominator of the right side makes this differential
equation non-linear
3) There is a sign
ambiguity
Faced with all
this situation Schrödinger got an idea to modify the
wave equation like this,
Ψ=A exp[i(px-Et)/ ħ]
where, i=√-1
By this assumption,
now we can leave time independent SE unchanged.
Now, E=( i ħ/ Ψ)(∂Ψ/∂t)
Substituting this
result in time independent equation, we get,
- ħ^2/2m(∂^2
Ψ /∂x^2) + V(x) Ψ
=( i ħ)(∂Ψ/∂t)
This equation is called as Schrödinger’s
Time-dependent equation.
Okay…okay…finally we got our equation....So,can we now take rest….????
Hmm...actually the answer is NO!!!!
Imagine....what will happen if I say in-spite of getting our equation, still there is a major mistake lying in this equation…..
Imagine....what will happen if I say in-spite of getting our equation, still there is a major mistake lying in this equation…..
Okay...for that you have to wait for my next blog.....my next blog will be on that mistake!!!
So, don’t forget to subscribe to our website and follow me on social media for updates…..
To know basics of quantum world, astronomy and space exploration you can check out my book "Through the wormhole" on amazon kindle.
You can directly talk with me on Instagram
To meet more physics Enthusiasts please join our Facebook Page
Also for latest updates of my posts join me on TwitterTo meet more physics Enthusiasts please join our Facebook Page
Thanks for reading.....
See you next time!!!
-Ratnadeep Das Choudhury
Founder and Writer of The Dynamic Frequency
0 Comments