Derivations

# Understanding Schrödinger’s Time Independent Equation!!! Corner-stone equation of quantum-mechanics....!!!

Till now we were discussing only the
theoretical part of the quantum mechanics…. But now it's time to understand the
maths behind all this drama in a simple way.

All starts with the classical wave
equations. It is the root cause of all the thought process regarding the
derivation of

**Schrödinger’s equations**….**∂^2y/∂t^2 = v^2∂^2y/∂x^2**

So, can we convert the classical wave
equation and modify it in such a way by which we can accommodate the
characteristics of quantum matter waves???

The simple answer is no. Strings
exhibit wave phenomenon as a whole; y and v in the above equation refer to
amplitude and speed respectively.

Erwin Schrödinger |

Schrödinger took assumptions that the
quantum wave equation should have two properties. As we have seen in classical
wave equation that the wave equation should have a linear and

**second-order differential equation**and the success of Bohr’s model argued for retaining the concept of energy conservation.
In classical physics, the conservation
of mechanical energy is expressed by,

**E= p^2/2m +V(x)**

**……….. (1)**

Remember, when we were discussing about
harmonic motion in elementary physics texts, we often encounters solutions of
the wave equation of the form

**y= Asin (kx**

**-**

**wt)**

**…...……. (2)**

Now,
one more equation we need to establish the relation between the classical wave
to quantum-mechanical matter waves, and that is the

**de Broglie relation**,

**λ**

**=h/p**

**…………..(3)**

Also, we know from wave mechanics that,

**k=2π/**

**λ**

**.………… (4)**

from equation 2,3 and 4,

**y=A sin {(p/**

**ħ)x – wt**

**}**

Now, double differentiating it with respect to x,

**p^2****=****- ħ^2/y(****∂^2y/∂x^2)****……………(5)**
Now, putting equation 5 in equation 1,

**-**

**ħ^2/2m(**

**∂^2y/∂x^2) + V(x)y=E y**

**…………….(6)**

Now, we can see that this equation is a

**non-relativistic**,**one-dimentional**,**time-independent Schr****ö****dinger’s equation**which is also known as Schrödinger’s Wave equation. Here, if you observe carefully then you can see that here there is no mention about the wavefunction.
This equation works fine for a pulse travelling on a string but when you
are taking about atoms and molecules then you should mention about 3-D space.

So, what Schrödinger did actually
to solve this problem????

By intuition equation 6 can be represented in a 3-D form by adding

**second partial derivatives of the wave pattern**with respect to y and z to the left side.
So, finally we get the 3-D form of Schrödinger’s
Time Independent Equation,

**-**

**ħ/2m(∂^2Ψ /∂x^2+∂^2Ψ /∂y^2+∂^2Ψ /∂z^2) + V Ψ =E Ψ**

But, here story doesn’t end…..

Anyone who knows the basic aspects of
physics must have noticed following curious aspects from the final equation:-

**1) Classical Wave equation was in fact never used. It only used as a prototype**

**wavefunction.**

**2) Nowhere the concept of quantization built into development. It only can be**

**seen by solving the equation for a set of values of V(x, y, z) and set of boundary**

**conditions**

**3) The total energy E appears explicitly in the final equation which is unusual for**

**classical physics as in Newton’s second law i.e. F=ma, there is no mention of E.**

**We can calculate that by providing boundary conditions like ignition position**

**and speed of the particle involved after solving the differential equation.**

For a given set of V(x, y, z) and set
of boundary conditions there exist certain values of Ψ(x, y, z) that satisfy the equation and each
values gives a particular value of E. This set of values of E represent the
quantized energy states of the system.

In my next
articles on quantum-verse, I am going to explain

**the need of time-dependent equation of Schr****ö****dinger and also going to derive it**. So, don’t forget to**subscribe to our website**and**f****ollow****me on social media**for updates…..*You can directly talk with me on*

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*Thanks for reading.....*

*See you next time!!!*

-Ratnadeep Das Choudhury

Founder and Writer of The Dynamic Frequency

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