QBasic: Schrodinger Wave Equation

In this Introductory article, we discuss about the derivation of the time in-dependent and time dependent Schrodinger wave equation.

Before starting would like to remember our unsung ancestors who contributed to the natural philosophy, over the time which reformed and extended as Modern physics.

Setup:

Classical physics describes the world in two very distinct ways.

  1. Objects behave as particles or
  2. Objects behave as waves.

The distinction as to which, is determined by certain characteristics displayed by the object.

For example, wave-like properties include wavelength, frequency, and interference patterns. Particles, on the other hand, are localized, and can be defined by position and momentum.

The classical interpretation of the world states that a body can act as either a particle or a wave. Never both.

Another aspect of classical physics is that properties of an object can have continuous values. In other words, characteristics of the object such as momentum and energy, can take any value. These values can be known to arbitrary accuracy. That is, it makes sense to talk about the position and momentum of an object with absolute certainty.

The discovery of Quantum physics has demonstrated that this Classical view of the world is not quite complete. As further investigations take place, it becomes more and more apparent that the above assertions of physics can no longer explain certain experimental results.

Young’s double slit experiment:

Young’s double slit experiment

Describes the wave like properties of light.

Huygens principle:

The principle can be used to predict the spreading of light wave. it is a geometrical construction using every point on the wave front as a source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the wave.

Huygens principle

waves interact with each other, producing an interference pattern with the minimum and maximum in the same areas.

When light waves are in phase creates constructive interference. when the light waves are out of phase, they create destructive interference as shown in fig: Young’s double slit experiment.

Photoelectric effect:

The photoelectric effect is a phenomenon in which electrons are ejected from the surface of a metal when light is incident on it. These ejected electrons are called photoelectrons. It is important to note that the emission of photoelectrons and the kinetic energy of the ejected photoelectrons is dependent on the frequency of the light that is incident on the metal’s surface.

Photoelectric effect
Planck–Einstein relation
  • E denotes the energy of the photon
  • h is Planck’s constant
  • f denotes the frequency of the light

De Broglie’s Equation:

De Broglie considered photons do have mass then

from Einstein's, E = mc².

De Broglie came up with an expression for the momentum of a wave as

we know,

using above conditions,

which gives the below condition,

So, the wave length 𝜆 of an electron which does have mass is as shown below,

here we replaced p with mv

the above condition gives us,

For an electron the mass is so small then it will have wavelength 10-¹⁰ 𝑚

so electrons behave as both particles and waves.

What kind of wave is an electron?

An electron is an circular standing wave but not a linear one. Circular standing waves can only have integer numbers of wavelengths.

There is a gap between energy levels because the electron can only have integer numbers of wavelengths which is the discrete level of energy. which tells Energy of an electron is quantized. Photon absorption promotes the electron to a higher energy level.

more wavelength = more Energy carried by the wave

Heisenberg Uncertainty Principle:

The principle states that the position and momentum of an object can never be known to 100% accuracy.

Heisenberg Uncertainty Principle

where,

Δ𝑥−uncertainty in position

Δ𝑝−uncertainty in momentum

ℏ is reduced Planck's constant

In short, the Heisenberg Uncertainty principle states the more we know about the position of an object, the less we know about the momentum of that object and vice versa.

Extracting from the formula, we can see that Δx and Δp are inverses of each other, or “conjugate variables”.

Schrodinger equation:

In Quantum mechanics, the information about the system is contained in the solution to the Schrodinger equation, where the square of the function is interpreted as the probability density. It tells us virtually everything we need to know about the system.

Let’s start with energy definition of a particle in classical physics,

the total energy of a particle is sum of it’s kinetic energy and potential energy

and replacing with basic definitions we discussed above give us as shown below,

From Quantum physics if we want to know the information about the electron we have to consider the wave function and find either momentum or position of an electron with respect to time or space.

energy equation

Here E is the total energy, V is the potential energy and Ψ(𝑥,𝑡) is the wave function.

Let us consider a complex plane wave function (Quantum wave functions are complex functions) which is a sum of cosine and sine wave forms,

complex plane wave function

Time Independent:

considering only space, which makes time-independent or constant time.

In terms of position, differentiated equation is defined by

first order,

second order,

third order,

similarly, nth order

from De Broglie relation

and k is a vector

where ℏ is reduced Planck's constant

from the above Energy definition(fig: energy equation) we can identify 𝑝²

to achieve 𝑝² we need 𝑘², so we consider second order partial differential and replace as shown below

i² as -1 and k² from the above condition,

re arranging the equation as shown below,

replacing the 𝑝²Ψ in the fig: energy equation, we get as shown below

or

Laplace Operator in 1-Dimensional,

Laplace Operator

if we consider x, y and z axis then it becomes 3-dimensional.

replacing in the above equation we get as shown below

taking wave equation as outer side

with Dirac notation,

Where,

H is an Hamiltonian Operator,

E is amount of Energy the electron is allowed to have.

This is a time-independent Schrodinger wave function. The meaning of this wave equation is that Hamiltonian operating on the wave function is to find the energy of the electron.

What if the total energy of the quantum mechanical particle is not constant in time?

Time Dependent:

for simplicity, we assume that particle is not in an external field and therefore has no potential energy.

with this assumption the total energy of the particle equals to only the time-dependent kinetic energy

total energy

from the time-independent energy equation

by replacing with the above condition in the (fig: total energy) equation we get as shown below,

total energy without P.E

from the complex plane wave function

partial differentiation with respect to time gives as shown below,

after re-arranging,

from fig: Planck–Einstein relation,

replace h with reduced h as shown below,

replace angular frequency as shown below,

replacing with angular frequency in the differential equation with respect to time as shown below,

re arranging the equation as shown below,

replacing the energy equation(fig: total energy without P.E) with the above terms, as shown below

This is time-dependent Schrodinger wave equation for a special case particle with out potential energy.

Let us consider the time dependent wave equation including the potential energy

time dependent wave equation

Solving the time dependent Schrodinger equation is not easy.

But we can simplify the solving of this partial differential equation by converting it in to two ordinary differential equation.

One differential equation depends only on time and the other only on space. the trick is called separation of variables.

Not all wave functions can be separated in this way. But since the Schrodinger equation is linear, we can form a linear combination of such solutions and thus obtain all wave functions(even those that cannot be separated)

performing space and time derivatives we get, as shown below

replacing this terms in fig: time dependent wave equation, we get as shown below

Since the function depends mainly on one variable we try to preserve the ordinary differential d in place of partial differential, as shown below

From the above expression we can observe that left hand side is time-dependent and right hand side is time-independent.

if time is the varying factor then right hand side is constant

Similarly, if space is the varying factor then left hand side will be constant

This is nothing but the time in-dependent Schrodinger wave equation

further exploring the time dependent part,

re arranging the dt, as shown below

integrating on both the sides

integral of 1/x is “ln x”, integral of dt is t and ignoring the constant

after re arranging the equation in to exponential

replacing the above term in the variable separation equation

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy.

where H is the Hamiltonian Operator.

From the above expression, we can understand that as the time evolves it rotates around the complex plain.

Ψ(𝑥,𝑡) is time dependent where as |Ψ(𝑥,𝑡)|² is not time dependent(complex conjugate get cancels or we can say ignoring the global phase) and |Ψ(𝑥,𝑡)|² is probability of observing an electron, collection of this probabilities with given energy is known as orbital.

Since, Schrodinger wave equation is linear, we can write a linear combination of the above result.

Reference: Wikipedia, stack exchange

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