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 Schrödinger’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 Schrödinger and also going to derive it. So, don’t forget to subscribe to our website and follow me on social media for updates…..

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See you next time!!!

                                                                    -Ratnadeep Das Choudhury

                                                      Founder and Writer of The Dynamic Frequency