IMPORTANT DERIVATIONS OF CHEMISTRY(SCHRODINGER WAVE EQUATION AND PARTICLE IN A BOX)

COURSE-B TECH
SUBJECT- CHEMISTRY
MOST IMPORTANT DERIVATIONS FOR
EXAMS, SCHRODINGER WAVE EQUATION
AND ENERGY OF PARTICLE IN ONE
DIMENSION(PARTICLE IN A BOX)
Derivation of Schrödinger's Equation:-


So, when we talk about the wave function, which is represented as

Ψ(x,t), it basically tells us the probability of finding an electron at a specific spot
(position 'x') and at a certain time ('t')1.


Now, for a wave function to be valid, it needs to have a few key characteristics:


●​ It should always be a single value.
●​ It has to be continuous.
●​ And, importantly, it must be well-defined.​


To derive the core of it, the Hamiltonian operator

H^ acting on the wave function Ψ(x,t) gives us the energy E multiplied by the wave
function Ψ(x,t)4.


The general wave equation looks like this:




∂x2∂2Ψ​(x,t)=V2VΨ​∂t2∂2Ψ(x,t)​5

, Here, 'V' is velocity, 'x' is position, and 't' is time.


We can separate the wave function into a spatial part and a time-dependent part:




Ψ(x,t)=Ψ(x)Ψ(t) 7




More specifically, the time-dependent part often involves a cosine function:




Ψ(x,t):Ψ(x)cosωt 8


And

ω is the angular frequency, which is 2πν9.


When we apply the second derivative with respect to x on the wave function and the second
derivative with respect to t, we get:




∂x2∂2​[Ψ(x)cosωt]=V21​∂t2∂2​[Ψ(x)cosωt] 10


This simplifies to:




cosωt∂x2∂2​Ψ(x)=V21​Ψ(x)∂t2∂2​(cosωt) 11


Which further reduces to:




cosωt∂x2∂2​Ψ(x)=V21​Ψ(x)[−ω2cosωt] 12