Schrodinger Wave Equation and Wave Function

The general one-dimensional Schrodinger Wave Equation is expressed as

Where, ψ(x, t) is the wave function, V(x) is the potential function and it is assumed to be independent of time. m is the mass of the particle and j is the imaginary number √(-1).
The wave function ψ(x, t) is used to describe the behavior of the system mathematically ψ(x, t) can be a complex quantity.
The wave function ψ(x, t) can be rewritten as

Where, ψ(x) is the function of position x and φ(t) is the function of time t.
Now the general form of Schrodinger Wave Equation can be rewritten as

Now left-hand side of the equation is only dependent upon the position x and right-hand side of the equation is dependent upon only time t. Each side of the equation must be equal to a constant quantity say η. This is because time-dependent side and position-dependent side of the equation is equal to each other.

Hence for the time-dependent side the equation would be,

Now, the solution is similar to classical exponential form of sinusoidal wave where η/(h/2π) = 2πη / h = ω = angular velocity of the sine wave.
Now as per quantum mechanics,

Separation constant of the equation is E, hence,

This is time independent form of Schrodinger Wave Equation.

Alternatively Establishing Time Independent Schrodinger Wave Equation

Now we will be trying to establish the time independent form Schrodinger Wave Equation and for that let us consider the wave equation

Now wavelength λ and momentum p of the wave are related to each other by the following equation called de Broglie wavelength equation,

Putting this value of λ in above second order differential equation we get,

Total energy of electron


Significance of Wave Function

We have already seen that the time and position dependent wave function ψ(x,t) can be rewritten as

As we have already proved, energy of electron E = η

In the year of 1926, Max Born stated that and postulated that if wave function of a particle is ψ(x, t), then probability of finding that particle between a gap of x and x + dx is,

Now we know some basic mathematics,

Therefore, |ψ(x,t)|2 can be written as,

Again as per basic complex mathematics,

Hence, it is proved that probability density function of a particle is independent of time. Hence, for finding position of electrons in crystal we should only concern with the time-independent wave function. It is needless to say that the probability of finding a particle anywhere in the universe is one. That means it must exist between the position – ∞ to + ∞. This convention then is mathematically represented in quantum mechanics or quantum physics by wave function as,

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