More Quantum Mechanics: The Schrodinger Equation

German physicist Erwin Schrodinger, developed the equation that goes by his name ca. 1926.


In this blog, we change gears from the political, environmental and religious issues and go back to physics, to look at more quantum mechanics (QM). Although I did examine the Schrodinger equation and its applications earlier: e.g. http://brane-space.blogspot.com/2011/01/solution-of-simple-quantum-mechanical.html, it helps to look into it at a more basic level, in terms of its own internal "mechanics" and fundamentals. This will incorporate some elements not covered in the more advanced treatment, including how the DE itself is obtained from first principles.

The Schrodinger equation was developed by Erwin Schrodinger around the same time (1926) that Werner Heisenberg developed his matrix wave mechanics. Both are ways to analyze quantum dynamics for assorted simple systems (square well, assorted barriers, 3D-box etc.) and processes, but the first caught on much more than the latter. Why? Probably the chief reason is that Schrodinger's version relies on plain old differential equations, as opposed to the more obscure matrices of Heisenberg. Thus, since most advanced physics students also have taken DEs, they are able to much more easily follow and arrive at the solutions. See, e.g.:

http://brane-space.blogspot.com/2010/06/looking-at-basic-differential-equations.html

And my other blogs to do with DEs:

http://brane-space.blogspot.com/2010/06/back-to-diferential-equations.html

http://brane-space.blogspot.com/2010/06/differential-equations-iii.html

http://brane-space.blogspot.com/2010/06/differential-equations-iv.html

http://brane-space.blogspot.com/2010/06/differential-equations-concluded.html


These will definitely help to refresh readers' perspectives before delving into this blog.

We begin by writing the general equation for a progressive wave U:

U = A exp{2πi(Kx - ft)}

where K the wave number is equal to 1/L (L = wavelength), x is the displacement (1-dimensional for simplicity), and f is the frequency. Meanwhile, for standing waves such as would be expected in many quantum applications:

U = A sin 2πKx exp(-2πift)

In either case (standing or traveling waves) U varies with x for a particular value of t in such a way as to satisfy the simple harmonic equation:

d^2U/dx^2 + 4π^2 K^2 U = 0

Which can be verified by direct differentiation (left to the industrious reader). If a field is present, so that the potential energy V(x) of an electron varies with x, K will be a function of x too. Based on the experiments from electron diffraction, one has:

K = p/h = m(e)v /h

where h is Planck's constant, and m(e) is the electron mass and v the electron velocity. Then:

K^2 = m(e)^2v^2/ h^2 = 2 m(e)(W - V)/ h^2

where W is the total energy of each electron so (W - V) is the kinetic energy, i.e.

W = V + [m(e)v^2/2] = V + KE

so: KE = W - V = [m(e)v^2/2], so:

2 m(e)(W - V) = [m(e)v^2]

Then: m(e)^2v^2/ h^2 = 2 m(e)(W - V)/ h^2

Now, substituting this last (for K^2) into the SHM equation, we obtain:

d^2U/dx^2 + 8π^2 m(e)/h^2 {W - V(x)}U = 0

And this is none other than one form (the 1D) of the Schrodinger equation.

Illustrating some basic properties of this DE is straightforward. The best approach is to apply it tos a specific case for which some parameters are known. Consider then an electron of charge e, moving in an electric field, E.

The electric force F(E) acting is: F(E) = eE

The potential energy V(x) is obtained from:

dV(x)/dx = F(E)

whence:

dV(x) = F(E)dx

and integrating both sides:

V(x) = F(E) x = eE x

The beauty of this is that the same argument can apply to any equation of the form:

d^2U/dx^2 + F(x) U = 0

where F(x) is some known function.

Another interesting facet of the Schrodinger equation refers to the superposition aspect.

If we start, say, with two different initial conditions, to obtain two waves:

U = U1(x)

and U = U2(x)

Then ALL solutions of the given Schrodinger wave equation are of the form:

U = A U1(x) + BU2(x)

Next: Determining specific values of the constants: A, and B for a specific simple system.