Physics-Schrödinger Equation

Schrödinger Equation

Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. before going to this eqyation, let’s talk some math related to this equation.

mathematics formulas

del operation:

\[\nabla = \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z}\hat{k}\]

Gradien of F: \(\nabla F = \frac{\partial F}{\partial x} \hat{i} + \frac{\partial F}{\partial y} \hat{j} + \frac{\partial F}{\partial z} \hat{k}\)

Divergence of F: \(\nabla \cdot F = (\frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k}) \cdot(F_1 \hat{i}+F_2\hat{j}+F_3 \hat{k}) = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}\)

Curl of F: \(\nabla \times F = (\frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k}) \times(F_1 \hat{i} + F_2 \hat{j} + F_3 \hat{k}) = (\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z}) \hat{i} + (\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x}) \hat{j} + (\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}) \hat{k}\)

or in the other way: \(\nabla \times F = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1 & F_2 & F_3 \end{vmatrix} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \hat{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \hat{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k}\)

Laplacian of A: \(\nabla ^2A = \nabla \cdot(\nabla A) = \frac{\partial ^2A}{\partial x^2} + \frac{\partial ^2A}{\partial y^2} + \frac{\partial ^2A}{\partial z^2}\)

Schrödinger Equation

This equation is about wave-function, nobody know what is exactly this wave-function in quantum physics but it can expalin many phenomenon in this subject and we use it a lot to explain some quantum properties.

Conceptually, the Schrödinger equation is the quantum counterpart of Newton’s second law in classical mechanics. (wikipedia)

How can you have a wave without wave equation?

Classic Wave:

$Y(x,t) = A.cos(Kx - Wt) \ \ \ \ \ \ @ K = \frac{2\pi}{\lambda}, W = 2\pi f$

So: \(\frac{\partial ^2 Y}{\partial x^2} = \frac{1}{V^2}.\frac{\partial ^2Y}{\partial t^2}\)

$V = f.\lambda \ \ \ \ \lambda =\frac{h}{P} \ \ \ \ \ f = \frac{E}{h}$

In quantum:

$E = E_k + E_p$

So:

\[i\hat h\frac{\partial \Psi(x,t)}{\partial t} = \frac{-\hat h^2}{2m} \frac{\partial ^2 \Psi(x,t)}{\partial x^2} + V(x,t) \Psi(x,t)\]

from this equation $\Psi(x,t)$ should be like thi:

\[\Psi(x,t) = A.e^{j(Kx - Wt)}\]

if $\Psi(x,t) = \Psi(x) \Psi(t)$ then we can write the equation Time Independent:

\[E \Psi(x) = \frac{-\hat h^2}{2m} \frac{\partial ^2 \Psi(x)}{\partial x^2} + V(x) \Psi(x)\]

Max Born

Probability of finding particle between x and x+dx:

\[P(x,t)\delta x = |\Psi(x,t)|^2 \delta x\] \[|\Psi(x,t)|^2 = \Psi ^*(x,t) \Psi(x,t)\]