# Quantum_Mechanics

## Schrödinger Equation

Schrödinger Equation: where $\hslash = \frac{h}{2\pi}$ is Plank's (revised) constant.

i\hslash = \frac{\partial \Psi}{\partial t} = - \frac{\hslash^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V\Psi

$\Psi(x, t)$ is a complex number because otherwise the equation is imaginary.

$\Psi(x, t)$ is a wave function that results a probability amplitude. Similar to Newton's second law, $\Psi(x, t)$ determines the "location" of object in future time given $\Psi(x, 0)$

The statistical interpretation: $|\Psi(x, t)|^2$ is the probability density of finding particle at point $x$ at time $t$.

\int_{-\infty}^\infty |\Psi(x, t)|^2 dx = 1
\int_a^b |\Psi(x, t)|^2 dx = \text{The probability of finding the particle between } a \text{ and } b \text{ at time } t

### Philosophical Consequence of Schrödinger Equation

Suppose I measure the position of the particle at point $x$, we ask where as the particle just before I made the measurement.

• Realist: "The particle was at $x$". This suggest quantum mechanics is an incomplete theory since particle has an exact position (nature is deterministic) but quantum mechanics is unable to tell as the exact position (because our ignorance). Hidden variable theory is needed to provide a complete description of the particle.

• Orthodox (Copenhagen Interpretation): "The particle wasn't really anywhere". The act of measurement disturb the particle and force it to produce an exact position.

• Agnostic (Wrong, eliminated by John Bell): Refuse to answer. Don't worry about things that can't be measured. Because you ask before, then we can't care about it. If you ask where the particle is at the measurement, then the particle is at $x$ by measurement.

• Many Worlds Interpretation: // TODO

Suppose I take another measurement immediately after the first measurement, would I get $x$ as the result?

• Yes: everybody agrees so. The measurement shrinks the wave function into a sharp peak around $x$. Wave function collapses.

### Solutions to Schrödinger Equation

Since Schrödinger Equation is a differential equation, if $\Psi$ is a solution, then $A\Psi$ is also a solution. So we impose the solution of Schrödinger Equation to sum to $1$. This process is called normalization.

\int_{-\infty}^\infty |\Psi(x, t)|^2 dx = 1

However, for non-normalizable (non-square-integrable) solutions (where $\Psi = 0 \lor \Psi = \pm \infty$), the solution can't represent particles (not physically realizable states) and must be rejected.

Now, Schrödinger Equation has property such that if we find constant $A$ that satisfy square-integrable along some specific time $t$, $A$ can also be used for other time $t$.

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