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equations:schroedinger_equation

$\color{royalblue}{i} \color{olive}{\hbar} \color{darkturquoise}{\partial_t} \color{firebrick}{\Psi(\vec{x},t)} = \color{darksalmon}{\hat H} \color{firebrick}{\Psi(\vec{x},t)}$

# Schrödinger Equation

## Intuitive

The Schrödinger equation describes how the state of a quantum system changes in time. It is as central to quantum mechanics as Newton’s laws are to classical mechanics

In quantum mechanics, we describe particles using waves. However, these waves are not waves in the same sense as sound waves or ocean waves. Rather, they are probability waves that tell us where it is likely to find a given particle.

Every time in physics we describe something in terms of waves, we need a wave equation that tells us how the waves behave in different situations. The Schrödinger equation is the correct equation to describe the probability waves of quantum mechanics.

The Schrödinger equation tells us that rate of change of the wave-function is completely determined by the total energy of the system.

In addition, it tells us that the wave-function is a complex function. Therefore, it cannot represent anything in our real world like, for example, a spatial displacement. Instead, to get something that gets actually measured in experiments, we must consider the absolute value of the wave function, which then represents the probability that something happens.

It also tells us that the actual size of quantum effects is tiny.

While in principle, Schrödinger's equation enables us to describe any kind of quantum system this is in practice quite hard. We can solve the Schrödinger equation only for very special simplified systems. For all other systems, approximation methods must be used.

## Concrete

In the Schrödinger equation, the wave-function $\color{firebrick}{\Psi(\vec{x},t)}$ describes the state of the system. This means concretely that ${ \Psi(\vec{x},t)}$ encodes, for example, if a given system is excited or in its rest state. The wave-function is a complex function, which means that at a given point in space, say $\vec{x}= (1,4,7)$, and time, say $t=9s$, the wave function is a complex number: ${\Psi(1,4,7,9)}= 5 + 3 i$. The wave-function at a given point in space and time yields a probability amplitude. By taking the absolute square of an probability amplitude we get the probability density and by integrating over some spatial volume we get the probability of the particle being within this volume.

The left-hand side of the Schrödinger equation denotes the of the wave function. We act with the partial derivative $\color{darkturquoise}{\partial_t}$ on our wave-function and the result: ${\partial_t}{ \Psi(\vec{x},t)}$ denotes how the wave function changes as time passes on.

The Hamiltonian operator $\color{darksalmon}{\hat H}$ represents the total energy of the system. We can get the Hamiltonian operator from the classical energy $E= T +V$, where $T$ is the kinetic energy and $V$ the potential energy by replacing

\begin{align} \text{ the classical momentum } p_i \ &\rightarrow \ {-i} \hbar \partial_{x_i} \, . \end{align}

Formulated differently, the Hamiltonian operator is calculated from the classical energy $E= T +V$ by replacing the classical momentum $p_i$ with the momentum operator $\hat{p}_i \equiv {-i} \hbar \partial_{x_i}$:

$$\hat H \equiv - \frac{\hbar^2}{2m} \nabla^2 + \hat V \hat{=} \frac{\hat{p}^2}{2m} + \hat V.$$

It is conventional to denote operators by an additional hat above the classical symbol.

The Hamiltonian is what is different for different systems. Formulated differently, the Hamiltonian characterizes the system in question. The rest of the Schrödinger equation stays the same for all systems. For example, the Hamiltonian for a harmonic oscillator reads

$$\hat H \equiv \frac{\hat{p}^2}{2m} - \frac{1}{2}k \hat{x}^2 .$$

Stationary Schrödinger equation

For systems with a time-independent Hamiltonian $\partial_t H =0$, it is helpful to split the wave function into two parts $\Psi(x,t) = \phi(t) \psi(x)$. We can see why this is useful by putting this ansatz into the Schrödinger equation

\begin{align} i \hbar \partial_t \Psi(x,t) &= H \Psi (x,t) \notag \\ i \hbar \partial_t\phi(t) \psi(x) &= H \phi(t) \psi(x) \notag \\ i \hbar \frac{\partial_t \phi(t)}{\phi(t)} &= \frac{H \psi(x)}{\psi(x)} . \end{align} Take note that Hamiltonian $H$ on the right-hand side contains the momentum operator $i \partial_{x}$ and thus $\psi(x)$ does not cancel here. The left-hand side now only depends on $t$ while the right-hand side only depends on $x$. This equation can only be correct for arbitrary values for $x$ and arbitrary values for $t$ when both sides independently yield some constant that we call $E$: \begin{align} \Rightarrow \quad i \hbar \frac{\partial_t \phi(t)}{\phi(t)} &= E , \Rightarrow\frac{H \psi(x)}{\psi(x)} &= E. \end{align}

The solution of the first equation is

$$\phi(t) = A e^{-Et/\hbar}$$ and the second equation is known as the stationary Schrödnger equation. This means that for all systems where the Hamiltonian does not explicitly depend on the time, we know immediately what the time-dependence of the total wave function $\Psi(x,t)$ looks like, namely: $\Psi(x,t) = \phi(t) \psi(x) = A e^{-Et/\hbar} \psi(x)$. The only thing we then have to do is to solve the stationary Schrödinger equation

$$\colorbox{lightpink}{H \psi(x)= E\psi(x)}$$ for the Hamiltonian $H$ that describes the system in question.

The complete wave function then reads

$$\colorbox{lightpink}{{\Psi(x,t) = \phi(t) \psi(x) = A e^{-Et/\hbar} \psi(x)}}$$

Example: Free Particle
The Hamiltonian for a free particle is $H= \frac{-\hbar \partial_x^2}{2m}$. The stationary Schrödinger equation therefore reads

\begin{align} H \psi(x)&= E\psi(x) \notag \\ \frac{-\hbar \partial_x^2}{2m} \psi(x) &=E\psi(x) \notag \end{align} This equation is solved for $E>0$ by

$$\Psi(x) =A e^{i\sqrt{2mE/\hbar}x} + B e^{-i\sqrt{2mE/\hbar}x}.$$

Such a solution describes a plane wave. However, since such a plane wave solution cannot be normalized (which we need since the total probability has to be 100\%), it is not a physical solution. Instead, to describe a real free particle we need to use a normalizable superposition of such plane wave solutions that we call wave packets.

Wave Packets

A generic solution of the stationary Schrödinger equation