Deriving the Schrodinger Equation

The Schrodinger Equation

The Schrodinger equation was formulated in 1925 and is used to describe how the state of a quantum mechanical system changes with time. The equation can be formulated with knowledge of differential calculus and of the quantum action principle.

The quantum action principle gives the following identity to a wavefunction of a free particle (Eq. 1):

Each point in spacetime has a wavefunction corresponding to a certain particle, and initially we will confine it to two dimensions: x and t.

Differentiating the wavefunction with respect to time yields a term for total energy (for a free particle, or when there is no potential energy on the particle), (Eq. 2):

Taking the second differential of the wavefunction with respect to space (in this case, just the x-dimension) yields a term for kinetic energy (Eq. 3):

It is assumed that energy is conserved, and a Hamiltonian from classical mechanics is used (the Hamiltonian is the sum of the kinetic and potential energy). It is also assumed that the potential does not vary with time, but it varies with space (which is why Eq. 2 resulted in the total energy, and Eq. 3 resulted in the kinetic energy).

The Hamiltonian is represented by Eq. 4:

Substituting the values of E and K from Eq. 2 and Eq. 3 respectively into Eq. 4 yields Eq. 5:

This can then be placed into the four dimensions of spacetime by using the del operator,  using a distance 'r' to show the spatial distance from a fixed point in the system, and allowing the potential energy of to change with time yields Eq. 6:

 

This shows that the equation for the wavefunction as shown in Eq.1 is valid for a non-relativistic, free, spinless particle as it has been shown to be equivalent to the Schrodinger equation (which deals with non-relativistic, spinless wavefunctions) when applied to the principle of conservation of energy.

QED

(Equation 1 adapted from 'The Quantum Universe: Anything That Can Happen Does Happen', Cox & Forshaw) - Recommended Read!

Pogson's Law

Pogson's Law

This post is one in three of a series on Astronomy. It determines the distances of galaxies by looking at their magnitude

Magnitude

Stellar brightness is based on a convention first devised by Hipparchus. The brightest stars that can be seen with the eye have a magnitude of 1.0; the faintest have a magnitude of 6.0. Due to the invention of the telescope, stars with negative apparent magnitudes can be seen. It is important for us to note that the more negative the apparent magnitude, the brighter a star appears.

This is for the following reason: the human eye perceives equal ratios of brightness at equal intervals. This means the difference in brightness of a 100W light bulb to a 200W light bulb would appear the same as the difference.

On the original scale, the flux (W/m^2) from stars of the first magnitude was about 100 times greater than that from stars of the sixth magnitude. Thus a difference of 5 magnitudes gives a flux ratio of 100. A magnitude difference of 1 would therefore correspond to a flux ratio of (100)^(1/5) = 2.512.

This can be re-formulated (as was by Pogson) into the mathematical definition: 

m(1) - m(2) = -2.5 log [f(2)/f(1)]

Where: m(1) and f(1), and m(2) and f(2) are the apparent magnitudes and fluxes of star 1 and star 2, respectively.

To estimate stellar and galactic distances, an absolute scale must be established. This can be done by looking at the consequences of having all stars equidistant from Earth, meaning the difference in stellar and galactic luminosity would be the only factor affecting the luminosity. The absolute magnitude, M, is therefore the apparent magnitude a star or galaxy as if it were 10 parsecs from Earth.

Assuming that the light is spread evenly by the star or galaxy in question, then the flux can be determined by the equation: f(10) = L/4(pi)(r^2), where r is 10 parsecs, and f(10) is the flux a star would have at that distance, and L is the luminosity of the star.

Thus: m-M = -2.5 log [f/f(10)], and f/f(10) = ((10^2)/(r^2)) = (10/r)^2

We can now re-write Pogson's law as: m-M = -2.5 log[(10/r)^2]
= -5 log(10) + 5 log r

Thus: m - M = 5 log r - 5

Which can be re-arranged to: r = 10^[(m+27)/5]

QED

Therefore, we can use the apparent magnitude of distant stars to show their distance from Earth in parsecs. For example, the galaxy coma2 has the apparent magnitude of 12.55 and thus is 81.28 Mega-parsecs from Earth.