The Uncertainty Principle and Schrodinger Equation

The theory of quantum mechanics is based on the probability of how things are probable to happen or how probable an event or particle is to occur or appear in the quantum regime of time and space. As the Heisenberg uncertainty principle states: The measurement of a particle will introduce disturbance to the system, which in fact will produce an error in the precision of the measurements we tend to take. There are complimentary variable in quantum mechanics like position and momentum, energy and time which are Fourier transforms of one another. It is by nature of quantum mechanics that measuring one of the conjugate variables precisely will affect the system of another variable which in turn will result in deviation of the other value. Thus the two pairs can never be simultaneously measured precisely. The mathematical equation follows as:

∆x.∆p≥h/4π

Where h represents the Plank’s constant.

Now if we take that this equation does represent the total error in measuring the position and momentum. We will take a case by case basis where each of the parameters will have a precise value measured and other will have a highest limitation on the values of, but the value will still lie in the probability range of (0–1) and will tend towards 0 but not equal to 0.

Now for a particle in whose wave function be represented by ψ along the x direction. Let it be moving in a one dimensional plane and whose trajectory is along the x axis, we will define the position at any point on the plane by co-ordinates of x.

Also let p represent its momentum along the x plane. Also consider it is a free particle and not subject to any external potential and moving along a constant potential V.

Definition of terms of time dependent wave equation

The Schrodinger wave equation for the particle is thus given by:

Ψ =
ⅆ²x/ⅆΨ²+2m/ℏ²(E−v)Ψ=0

For x = 0, the solution of the equation is given by

Ψ= Acos𝛂x+Bsin𝛂x

For the other part of the equation, when x is not equal to 0,

The solution of the equation is given by:

Ψ =Ce^𝛃x ++ De^−𝛃x

Until the value of x does not tend to infinity, x will be defined by values of C and D and tend to change exponentially when the value of x is finite.

Solving the above two equations we get that the wavefunction vanishes at the surface of the infinite potential well. Now, the value of the wavefunction must satisfy the following conditions:

must be finite when integrated and

1. wavefunction must be continuous and single valued

2.must be continuous and finite.

Now we will discuss the variations in the wavefunction, if the value measured of x has an error associated with it.

The maximum value of error that can be gained is the original maximum range value of x itself. If the perimeters are set as such the range lies between some finite values the maximum value that x will have an error will be 1. This implies that the probability which is defined in terms of the Ψ functions will have a different value. The value of < Ψ2> thus will have a probability measure in terms of the maximum uncertainty in a region the particle can be found.

This also implies that if we add a range of error the exponential term of the wave equation, it will represent the region where the particle will be least likely to be found since x denotes a point in the plane where the particle is located.

This will further imply that that the value of {1-< Ψ2>} should thus give the probability in regions of space where the particle cannot be found. As solutions for the calculations of the wavefunctions still exist in the vicinity of regions where the uncertainty of finding the particle is maximum, the solutions should exist in regions of space before even the wavefunction in the given regions of space even evolved. Thus a particle should have probabilities of existence in regions where the range of measurement or the range of the maximum value of space it was never even confined to.

If we apply similar concepts to the momentum of the particle , it ought to represent the same results , where the momentum measurement will be less likely to provide the existence of the particle.

Thus as a result of this we can identify that the process of the errors will not always occur due to the error an observer makes but due to the fundamentals of the quantum mechanics itself. As quantum mechanics states only a finite region sought be considered where the particle is to be found, but the particle is always there on the other side of the negative probability region where we don’t look for it. Thus the uncertainty principle will not only locate a particle in the probable regions of the wavefunction, but due to the nature of the wavefunction it is possible that the particle may be on the other remaining ranges of the wavefunction at the similar time. This extends the probability regions for the wavefunctions, although small but these can be the origin of the quantum tunneling effect.

The process that we generally encounter as tunneling in quantum mechanics is an effect where a particle of lower energy will still have possibilities of emerging on the other region of the high potential barrier(V > E). It is supposed that the particle will burrow energy from its surrounding to cross the potential barrier. The exact origins of the tunneling has yet not been researched for properly or described.

Here , with the help of simple arguments, we thus proved that the probability that a particle can be basically found within the given limits exits and wavefunction still satisfies the other conditions. With the arguments we can argue that the quantum particle is in duet state where one of the probability regions would be the one where it will most likely to be found.

Thus , the process of tunneling can have a origin in the basic explanation of the wavefunction itself. It should be viewed as a nature of quantum mechanics itself.