[QuantumClue (OP)]

Mods I was in two minds whether to just post this in the Dirac Equation thread. If you want to merge that is absolutely fine with me.


The Schordinger Equation

THE Schrodinger equation is probably one of the most recognizable in physics today, alonside perhaps Eulers theorem and the Dirac Equation.  Almost anyone who has had a small interest in physics comes to learn that matter has a double description known as the particle-wave duality, or has been called by some, the wavicle. This dual representation of matter grows from the soil of experimental physics. The wave function as a probability description yields a most interesting nature for matter. Indeed as the name suggests, somehow matter not only exhibits a particle nature, but it also exhibits some kind of wave nature.

The wave nature comes in mathematical jargon as a wave function Ψ which dictates the likelihood of finding wave distributions, where the probability is found as the absolute square of the wave function P=∫|Ψ|².

Deriving the wave equation governing all matter, called the Schrodinger equation is quite easy.

A wave function which depends on time and position is given by:

Ψ(x,t)=e^i(kx-ωt)

This function should be recognizable. We have used it in the previous writeup on the Dirac Equation, except this time our wave function depends both on position and time. I'm now going to show off and talk a tiny bit about the mathematical relationships. These are linear relationships, which I personally feel are much easy to understand.

A state vector in Dirac Notation is given as

|ψ(t)>

And here specifically it is said to be a function of time. To every state vector there is postulated to be a dual vector space:

<ψ(t)|

and this is the complex conjugate of |ψ(t)>. In notation, we can state that (|ψ(t)> ξ R) where ξ will be our symbol notation for ''in'' ~ which refers to a set. So we translate (|ψ(t)> ξ R) as saying |ψ(t)> is in the real number range R. So  (<ψ(t)| ξ C) says its vector <ψ(t)| is in the complex range C.

Such vectors can have very simple descriptions like:

|ψ(t)>=a+ib

<ψ(t)|=a-ib

where (a) is the real part and (ib) is the imaginary. Mathematically if position depended on time and the wave function was time dependant then:

ψ(x,t)=ζ_x ψ(t)

We have our usual mathematical notations we would find when studying linear algebra:

Orthonormal Basis and Basis Vectors in a Hilbert Space

|ψ>=Σ^∞_n=1 c_n|φ_n>

Where our Orthonormal Basis is |φ_n>. The inversion of this is the formula:

c_n=<φ_n|ψ>

finite norm

<ψ|ψ>=Σ^∞_n=1|ψ_n|²=||ψ||²

Going back now to our recognizable function:

Ψ(x,t)=e^i(kx-ωt)

We realize soon that pulling the derivate with respect to (t) gives us a (-iω). Writing this is quite easy into an equation:

(∂/∂t)ψ=-iωψ

After this we can see this wave equation simply transforms through very elementary derivision to:

Hψ=ih(∂/∂t)ψ

Where (H) is the Hamiltonian, which is simply the total energy of the system. A three dimensional case of this equation would have the form:

ih(∂/∂t)ψ=E▼²ψ+Vψ

Where ▼² is the three dimensional laplace operator. In Cartesian Coordinates, this wave equation is expressed as (∂²/∂x²)+(∂²/∂y²)+(∂²/∂z²).

Time Evolution

A system will evolve due to the schrodinger equation, which in its abstract form has been presented as:

Hψ=ih(∂/∂t)ψ

We find the relationship

H|Φ_n>=E|Φ_n>

Where |Φ_n> are some group of orthogonal vectors which span our space.

The reason why I introduced the identity of a wave function dependant on time becomes clear now. It turns out that it depends on time as it is spanned throughout the space:

|ψ_n(t)>=e<sup>iω_nt|Φ_n>

It is probably best to note, that this equation is not strictly linear in all cases.

Relativistic Schrodinger Equation

The Schrodinger equation has taken a form so far which is both linear and non-relativistic. This means the equation does not take relativistic effects into consideration. This is acheived by a new derivation, which involved the Klein Gorden equation. To create the Klein Gorden equation (or as Foolosophy has complained, The Klein-Fock Gorden equation), we have a reasonably easy derivation to produce a quantized equation. Quantization just means in this sense that we will replace all known variables with their respective operations.

We first begin with the Kinetic energy equation,

KE=1/2 Mv2

which is the same as the expression p2/2M where p should be replaced by the momentum operator p=i/h▼. Through a simple derivation of applying a statistical nature and the energy operator ih(∂/∂t)ψ we have the non-relativistic case of our Schrodinger Equation, which is easy to show if you replace the operations into a single equation.

Turning our attention to the energy-momentum relationship:

E2=p2c2+M2c4

which when quantized via replacement of operators gives:

ih(∂/∂t)ψ=(√(i/h▼c)2+M2c4)ψ

Removing the imaginary numbers, and replacing the wave part with the differential laplacian gives us our relativistic Schordinger equation:

▼2ψ-1/c2 ∂2/∂t2 ψ=M2c4/h2ψ

The solution to this equation is our well-known function Ψ=ei(kx-ωt). It's use in these two last writeups have been paramount.</sup>

[QuantumClue (OP)]

There where a few typos there in the math, but its all fixed now.