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!!!**

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-Ratnadeep Das Choudhury

Founder and Writer of The Dynamic Frequency

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