DeBroglie's matter waves weighing heavy upon his mind, Erwin Schrodinger wanted time to ponder, time to consider all the implications. Schrodinger, an Austrian physicist noted for his work on the physics of strings, took flight to a villa in the Swiss Alps in 1925, leaving his wife behind and gathering a former Viennese girlfriend. What would come of this (presumably) quiet period of reflection and thought would forever change the landscape of physics. Indeed, it would change the way we as a species reckons the universe we live in.
A sort of microscopic solar system, with electrons orbiting about the nucleus like planets to stars -- the Bohr atomic model was proving to be of limited utility. For hydrogen atoms, the agreement between predicted and observed behavior was sterling. However, for atoms with more than one electron -- even helium with only two electrons -- predicted and observed behavior radically diverged. Schrodinger desired to develop a model that agreed with the experimental evidence. What came of that illicit vacation to the Swiss Alps was a model that was not derived from any other, a model that can be called an intuitive guess, a leap of imagination, a model that is astonishingly accurate.
In these pages, we will, for the sake of both brevity and simplicity, only consider the time-independent Schrodinger wave equation in one dimension. We will not consider the full equation in all of its gruesome splendor. The time-independent Schrodinger Wave Equation, which could validly be called Schrodinger's law, is given by the differential equation
where j (x) is the is the wave function, m is mass, is Planck's constant divided by 2p, E is the total energy of the particle, and U(x) is the potential energy function of the particle. As when one ingests something disagreeable and the natural reaction is nausea, so too is the natural reaction to this equation. However, comfort may be taken if we consider that acceleration is the second derivative of the position function and, therefore, could be written
As surely as acceleration simplifies to something more palatable, The Schrodinger wave equation must simplify (a little, at least).
To find general solutions to this equation, boundary conditions must be established. The principle conditions that it must adhere to are
|2. if x is in someplace it is physically impossible to be|
|3. j (x) is a continuos function|
|4. j (x) is a normalized function|
In these pages, we will, again for brevity and simplicity, consider the case of a particle in a one-dimensional box of ideal rigidity, such that its walls are impenetrable. Let the box have length L. As may be seen in the illustration, the potential
energy function has two states:
|1. U(x)=0 for|
|2. U(x)= for x<0 or x>L|
Since it is physically impossible for the particle to be outside of the box, it is the first state that is of interest. Indeed, this simplifies the wave equation considerably, with the term U(x) dropping out. Therefore, the wave equation corresponding to the particle in the box is given by
Before assailing this equation with a display of mathematical acumen, let us ask ourselves what function's second derivative is merely some negative constant -- all of the terms on the right-hand side save j (x) -- multiple of itself? To simplify, let
Therefore, the wave equation becomes
It becomes clear, a trigonometric function like sine or cosine would be a good candidate for j (x). Therefore, our guess for the solutions to the wave equations is
By the first above boundary condition, it is known that
where n=1,2,3,... When the smoke clears, we have that
where A is the function's amplitude. To determine the amplitude, recall the fourth boundary condition, j (x) is a normalized function. Mathematically, this means
In words, this states that the probability of finding the particle somewhere on the x-axis is one or 100%. Waving hands a bit to omit the gory details, this gives
Gasping for breath, we at last have unearthed the solution to the wave equation for the particle of the nth quantum state in the rigid box.
|for x<0 or x>L|
The utility of this solution lies primarily in that the probability of finding the particle at some position x is given by the square of j (x).
The importance of this relationship is best illustrated graphically. Consider a particle in the third quantum state.
It can be seen that there are regions where the probability of finding the particle is zero -- so-called nodes. This is not unique to simply the particle in a rigid box model. It is observed in more sophisticated ones such as the model of an electron orbiting a nucleus...
shrodinger wave equation is inspire by william hemilton's wave equations produced on the imeginary planet surarounded by sea.it like this
Schrodinger was the first person to write down such a wave equation. Much discussion then centred on what the equation meant. The eigenvalues (solutions) of the wave equation were shown to be equal to the energy levels of the quantum mechanical system, and the best test of the equation was when it was used to solve for the energy levels of the Hydrogen atom, and the energy levels were found to be in accord with Rydberg's Law.
It was initially much less obvious what the wavefunction of the equation was. After much debate, the wavefunction is now accepted to be a probability distribution. The Schrodinger equation is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wavefunction gives the probability of finding the particle at a certain position.
The Shrodinger equation is:
The solution to this equation is a wave that describes the quantum aspects of a system. However, physically interpreting the wave is one of the main philosophical problems of quantum mechanics.
The solution to the equation is based on the method of Eigen Values devised by Fourier. This is where any mathematical function is expressed as the sum of an infinite series of other periodic functions. The trick is to find the correct functions that have the right amplitudes so that when added together by superposition they give the desired solution.
So, the solution to Schrondinger's equation, the wave function for the system, was replaced by the wave functions of the individual series, natural harmonics of each other, an infinite series. Shrodinger has discovered that the replacement waves described the individual states of the quantum system and their amplitudes gave the relative importance of that state to the whole system.
Schrodinger's equation shows all of the wave like properties of matter and was one of greatest achievements of 20th century science.
It is used in physics and most of chemistry to deal with problems about the atomic structure of matter. It is an extremely powerful mathematical tool and the whole basis of wave mechanics.
The Schrodinger equation is the name of the basic non-relativistic wave equation used in one version of quantum mechanics to describe the behaviour of a particle in a field of force. There is the time dependant equation used for describing progressive waves, applicable to the motion of free particles. And the time independent form of this equation used for describing standing waves.
Schrodinger?s time-independent equation can be solved analytically for a number of simple systems. The time-dependant equation is of the first order in time but of the second order with respect to the co-ordinates, hence it is not consistent with relativity. The solutions for bound systems give three quantum numbers, corresponding to three co-ordinates, and an approximate relativistic correction is possible by including fourth spin quantum number.
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