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 Schrödinger Wave Equation

Austrian physicist Erwin Schrödinger lays the foundations of quantum wave mechanics. In a series papers he describes his partial differential equation that is the basic equation of quantum mechanics and bears the same relation to the mechanics of the atom as Newton's equations of motion bear to planetary astronomy.

The equation-

$H\Psi = E\Psi\,$

The mathematical description of the electrons is given by a wave function, , (or a State Function), which specifies the amplitude of the electron at any point in space and time. Wave functions are solution of Schroedinger's equation. For a one-dimensional particle, the time-dependent Schroedinger equation can be written,
 (a) 1s electrons can be "found" anywhere in this solid sphere, centered on the nucleus.(b) The electron density map plots the points where electrons could be. The higher density of dots indicates the physical location in which the electron cloud is most dense.(c) Electron density (Y2) is shown as a function of distance from the nucleus (r) as a graphical representation of the same data used to generate figure b.(d) The total probability of finding an electron is plotted as a function of distance from the nucleus (r).

Shape of Schrödinger Atoms

Schrödinger’s equation requires calculus and is very difficult to solve, but the solution of the equation, when treated properly, gives not the exact position of the electron (remember Heisenberg), but the probability of finding the electron in a specific place around the nucleus.  This most probable “place” is known as an orbital.  An orbital is a volume space around the nucleus that contains the electron 90% of the time.  Realize this space is determined from the solution of an equation and not from direct observation.  Also, it does not describe an orbit.  An orbital is very different, but the concept of an orbit as being a fixed distance (and therefore a fixed energy) from the nucleus will help us understand the idea of an orbital.  To get an idea of what an orbital is, picture a string hanging from the ceiling in a dark room.  The string has a cage of sex attractant attached to the end.  A firefly is let in and its light can be seen periodically.  If we record each flash of light on film over the period of a few hours, what would you expect to see on the film?  Where would most of the flashes be?  Most would be very close to the sex attractant but we would also see some flashes farther away, decreasing in number the farther the firefly got from the sex attractant.  If we could look at the multitude of flashes caught on film in 3 dimensions, we would see a sphere of flashes with greater density close to the center of the sphere.

This is very much like the possible positions of the electron in an orbital.  Most of the time, the negative electron will be close to the positive nucleus, but sometimes, it will not.  We cannot tell anything about when the electron (firefly) occupied a certain point, but looking at the whole volume of probability (orbital) we can see where it is likely to be found.  If we draw a circle around 90% of the flashes, we have defined one type of orbital, in this case, a sphere.  This shape would be one solution to the Schrodinger equation for where to find the electron in a hydrogen atom.   Recall in Bohr’s model that each electron orbit had a certain energy associated with it and only certain orbits were allowed, thus the energy levels of the hydrogen atom were quantized.  We need to set up this same idea of quantization for the orbital model of the hydrogen atom.  To do this, we will define principal energy levels and label them with integers n=1,2,3……  In the Bohr model, the larger “n” gets, the farther away the electron is from the nucleus and the greater energy it has.           Where   “n” = 1, the electron is in the ground state.  In an orbital, the principal energy level increases as the electron moves farther from the nucleus as well, but it moves from one orbital to another not from one orbit to another.  In the case of hydrogen, when the electron gains the right amount of energy, it moves to an orbital with principal energy level 2.  The principal energy level defines the energy (and therefore the average distance from the nucleus) of the electron, but it does not specify the shape of the probability volume (the orbital).  This shape is defined by the energy sublevel and is also a consequence of Schrodinger’s equation.  Each principal energy level in an atom has the same number of sublevels as its principal energy level integer.  The following table summarizes these relationships:

ModifiedFrom-http://www2.asd.k12.ak.us/hauser/curriculum/html/Chemistry/Unit%209%20Modern%20Atomic%20Theory/Handouts%20and%20Notes/Unit_09_Light_(Handout).htm

Bonus

Schrödinger's cat

Schrödinger's Cat: When the nucleus decays, the Geiger counter  may sense it and trigger the release of the gas. In one hour, there is a 50% chance that the nucleus will decay, the gas will be released and the cat killed.

from wiki-http://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat

/ Heisenberg / Planck / Schrödinger / Chadwick