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Schrödinger’s equation

In classical mechanics, Newton’s second law can predict the future behaviour of a dynamic system. Similarly, in quantum mechanics, given a set of initial parameters, the Schrödinger equation describes the evolution of a quantum system. The equation is written in terms of the wavefunction and can be used to analyse and predict the probability of events or their outcome. Before we jump right in to the Schrödinger equation, we should better understand the wavefunction.

Every particle can be represented by a wavefunction, typically denoted by the symbol $\psi$ .The wavefunction contains all measurable information of the particle it represents and gives us a probability distribution for the location of the particle. Simply put, it tells us where the particle is likely to be. Particles exists in a superposition of all possible places at the same time. Maybe you have heard of Schrödinger’s cat - we don’t know if the cat is dead or alive until we open the box. Similarly, we don’t know what state a particle is in until we measure it. Not knowing where the particle is, allows its probability distribution to be spread out like a wave. Once we measure the particle, it is forced to choose a state and the wavefunction collapses.

Important properties of the wavefunction

Satisfying the Schrödinger equation: The wavefunction must serve as a valid solution to the Schrödinger equation, which describes the system's quantum behavior.

Probability density and normalisation: The quantity $\psi \psi$ gives us the probability density - how likely we are to find the particle at a particular position. The larger the value of $\psi \psi$ at a specific point, the higher the probability of finding the particle at that location.

If the particle does exist, then when we sum $\psi \psi$ over all space, it must be equal to 1:

\int \left ( |\psi|^2 \right ) dV= 1

This equation is the normalisation condition. It ensures that the wavefunction is scaled such that the probability of finding a particle is 100% (1). In order to normalise a wavefunction, you often need to multiply it by a constant, C, to ensure the total probability equals 1 when the squared magnitude of the normalized wavefunction is integrated over all space:

\int \left ( |C\psi|^2 \right ) dV= 1

Continuity and smoothness: The wavefunction $\psi$ must be a continuous, well defined function of position $x$ and time $t$ without any jumps or discontinuities. This requirement is essential to ensure the mathematical and physical consistency of quantum mechanics. It enables the precise description of quantum systems, the conservation of probability, and the proper functioning of mathematical operators and equations in quantum theory.

Expectation value and variance: The expectation value, also known as the effective average value, provides a way to calculate the average outcome of a quantum observable (such as position, momentum, energy, etc.) for a given quantum state described by the wavefunction.

The expectation value, denoted as $\langle A \rangle$ , of an observable A is calculated as the integral of the product of the wavefunction and the operator corresponding to the observable:

\langle A \rangle = \int \psi^* A \psi dx

Here, $\psi^*$ represents the complex conjugate of the wavefunction $\psi$ , $A$ is the operator corresponding to the observable, and the integral is taken over all possible values of position $x$ . It should be noted that the expectation value can also be a function of time, $t,$ telling us how the average value of the obserable changes as the system evolves.

In the physical sense, the expectation value is just the average value for that observable. For example, if we consider the position operator, $X$ , calculating $\langle X \rangle$ for a specific wavefunction $\psi$ , would tell us where on average you would find the particle if you were to perform position measurements on many identical quantum systems in that state $\psi$ .

We can use the expectation value to calculate the variance:

\left (\Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2

The variance tells us how much the individual measurements of the observable $A$ are likely to fluctuate around the expectation value.

Free Particle Behaviour: A free particle refers to a particle that is not subject to any external forces or potentials. The wavefunction for a free particle is a sine wave, described as:

\psi (x,t) = Ae^{\left(ikx - i \omega t\right)}

Where $A$ is a normalisation constant, $k$ is the wave vector related to momentum and $\omega$ is the angular frequency. This wavefunction represents a particle with a definite momentum $p = \hbar k$ and totally uncertain position (due to Heisenberg’s uncertainty principle). The value $\hbar$ is the reduced Planck’s constant, $\hbar = \dfrac{h}{2\pi}$ .

The concept of free particles are convenient as a simplification to study the behaviour of particles in ideal conditions. In more realistic situations, particles are typically described as wave packets. A wave packet is a superposition of multiple sine waves with different momenta, allowing for a more localised description of the particle’s position.

Understanding Schrödinger’s equation

In the classical world, we can describe a system through Newton’s second law:

\text{total energy = kinetic energy + potential energy}

E =\dfrac{1}{2}mv^2 + \dfrac{1}{2}kx^2

writing this in terms of momentum, $p$ :

E=\dfrac{p^2}{2m} + \dfrac{1}{2}kx^2

Schrödinger’s equation is given by:

E \psi = H \psi

Note, this is the time independent version of Schrödinger’s equation. I’ll update this page soon to include the time dependent version! $H$ is the hamiltonian operator and $E$ represents the energy eigenvalues of the system.

In the case where we are dealing with a quantum harmonic oscillator, we can write Schrödinger’s equation in the form:

E \psi(x) = \dfrac{-\hbar^2}{2m}\dfrac{d^2 \psi(x)}{dx^2} + V\psi(x)

Here, E represents the energy the electron is allowed to have. If we solve the RHS of this equation then we can find the energy levels of the wavefunctions of the electron. With this, we can find out everything else we want to know about the particle. Comparing this form of Schrödinger’s equation to Newton’s second law, they look quite similar and we can actually derive the KE term using the de Broglie relation.

The solutions to the time independent Schrödinger equation take the form:

\begin{aligned} \psi(x) &= \sqrt{\dfrac{2}{L}cos \left( \dfrac{\pi n x}{L}\right)} \hspace{2cm}\text{for n = 1,3,5...} \\ \psi(x) &= \sqrt{\dfrac{2}{L}sin \left( \dfrac{\pi n x}{L}\right)} \hspace{2cm}\text{for n = 2,4,6...} \\ E &= \dfrac{\hbar^2n^2 \pi^2}{2mL^2} \end{aligned}

where $m$ is the mass of the particle, $L$ is length and $n$ denotes the energetic state. The full derivation of these solutions is quite long but I will add it to this page at a later date, along with example problems.

References

Hyperphysics - quantum physics

Wavefunction wiki

Schrödinger equation wiki