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.

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

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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