Wave Function

For hundreds of years, the scientific community perceived physics as the most precise science capable of plotting the motion of the planets and explaining the thermodynamics needed to power engines and machines. Who would have foreseen that Max Planck's innocent 1900 proposal that radiation consists of discrete energy packets would completely overturn humanity's presumed understanding of microphysics? Indeed, the dawn of quantum mechanics was quite a shock as physicists began to discover that nature was far from precise. In fact, probability and uncertainty were fundamental features of nature and not just reflections of our ignorance! At the heart of this probabilistic theory is the wave function which gives the most complete description of a quantum system possible. In this article, we will look deeper at the wave function and how we can visualize quantum systems. In doing so we'll come across one of the most intriguing and controversial questions in modern physics, the infamous collapse of the wave function.

Wave Function Definition

Quantum mechanics defines the state of a system probabilistically. This means that one cannot know the precise state of a system before making a measurement. Mathematically, a quantum wave function denoted by \(\Psi\) encodes these probabilities. This quantum wave function is a function of the degrees of freedom defining the possible states of the system. The wave function then outputs a complex number, known as the probability amplitude, whose modulus squared gives the probability density of the system being in that particular state.

Fig. 1. Image showing the probability densities of an electron around a nucleus in a hydrogen atom. The lighter the color the more likely the electron will be found there. This probability density is defined by the wavefunction \(\Psi\).

The energy level or orbital of the electron within the atom determines the shape of the probability cloud. Figure one shows the orbitals of an electron within a hydrogen atom. The electron is most likely to be found in the lighter regions whilst it will never be found in the black regions. In addition, the so-called quantum numbers characterize the shape of the orbital by defining the angular momentum and spin of the electron. At low energies, the orbital has a ring-like shape similar to the Bohr Model of circular orbits. However, at higher energies, the shape of these orbitals becomes more complex. This wave function model for atomic electrons is the most accurate picture of the atom we have and has been key to developing our theory of chemical bonding. Consequently, one of quantum mechanics' most important applications is its prediction of the properties of the periodic table.

Wave Function Graph

We can visualize the behavior of a quantum particle with the graph of its wave function. By looking at the graph's amplitude, we can see which regions are most likely to contain the particle. Charting how the graph changes over time also indicates how the quantum particle evolves. For example, look at the graph below representing the wave function of a particle in a box. The wave function is a standing wave with four nodes and three anti-nodes. The nodes of the particle's wave function are the regions where the probability of finding the particle are lowest, whereas the anti-nodes are regions where the probability to find the particle is highest.

Fig.2- A graph of the wave function of a particle trapped in a box. The standing wave formed shows that there are zones of high probability and areas of zero probability, much like the electron orbitals in an atom.

We can see this probability distribution more directly by graphing the absolute value squared of the wave function

\[|\Psi(x)|^2\]

The figure below shows the result. In this figure, the three clear peaks correspond to the anti-nodes of the standing wave.

Fig. 3. Plotting \(|\Psi(x)|^2\) gives the probability density distribution of the particles position.

The graph of \(|\Psi(x)|^2\) is a continuous probability distribution giving the probability density of the particle at each point. The area under the graph in some region \({a<x<b}\) gives the probability of finding the particle. This is equivalent to the usual procedure of summing probabilities in a discrete probability distribution. A larger area means that the probability of finding the particle there is higher. As we would expect, increasing the size of the region within which we look for the particle means a larger area under the graph and a greater chance of finding the particle.

Normalization of the Wave Function

We can place some key constraints on a wave function to ensure it describes physical particles in agreement with experiments. Consider searching for a particle within a box. Obviously, the particle must be somewhere if we look over the entire region of the box. Mathematically, we say that the sum over the probabilities for each position must always be \(1\). As we saw in the previous section, the area under the graph of \(|\Psi(x)|^2\) within some region is equivalent to this sum over probabilities. It follows then that the total area under the \(|\Psi(x)|^2\) graph must always be one. Note that this applies to any wave function. Even if \(\Psi(x)\) extends over an infinite space (as it does for free particles), the area under the \(|\Psi(x)|^2\) graph must always equal one.

This may seem like a massive constraint on the number of physically permissible wave functions. However, it is possible to find a physically permissible wave function from almost any wave function by multiplying it by a suitable factor, known as a normalization factor. Consider a graph of \(|\Psi(x)|^2\) , denoted \(A\) for some wave function \(\Psi(x)\). If the area under the curve is not equal to one, we can multiply \(\Psi(x)\) by \(\frac{1}{A}\) to ensure that the new wave function is physical.\[\Psi_N(x)=\frac{1}{A}\Psi(x).\]

Such a process is called normalizing the wave function \(\Psi(x)\). Incredibly, it turns out that the physics described by the normalized wave function \(\Psi_N(x)\) is entirely equivalent to the original un-normalized wave function. However, this normalization procedure only works for wave functions where \(0<A<\infty\).

Wave Function Collapse

It is important to keep in mind that the wave function of a system determines the probability of finding the system in some state or position upon measurement. However, something strange happens once we do make that measurement. Let's say we wanted to check the results of our first measurement by making a second measurement shortly after. For the result of our first measurement to be valid, this second measurement must return the same result. Otherwise, there is nothing to confirm our initial measurement. This means that once measured, a quantum system must remain in the same state. This has a profound implication for our understanding of the wave function.

Consider measuring the position of a particle, again described by some wave function. Initially, the wave function is a spread of probability-amplitudes over space. However, once we make a measurement we know for certain the position of the particle. Let's call this position \(C\). Any further measurement must always return \(C\), thereby making the probability at \(C\) one and zero everywhere else. This means the wave function becomes a sharp peak centered at \(C\) with zero amplitude everywhere else.

Fig. 5. After measuring the particle within the box, its wave function collapses down to a single point shown above as a sharp peak around \(x=1\). The probability of finding the particle at \(x=1\) is now \(1\).

We say that the measurement has 'collapsed' the wave function down to a single point. This collapse of the wave function demonstrates the mysterious nature of measurements in quantum mechanics. It is one of the strangest and most controversial aspects of quantum physics. The physics behind such a collapse, and whether such a physical interpretation is even possible, is still a topic of fierce debate.