Wave Functions: Definition, Properties, Equation & Signs

Richard Feynman once said, "If you think you understand quantum mechanics, you don't understand quantum mechanics." While he was undoubtedly being slightly glib, there is definitely truth to his statement. Quantum mechanics is a challenging subject even for the most advanced physicists.

The subject is so powerfully not intuitive that there isn't really much hope of understanding ​why​ nature behaves the way it does at the quantum level. However, there is good news for physics students hoping to be able to pass classes in quantum mechanics. The wave function and the Schrodinger equation are undeniably useful tools for describing and predicting what will happen in most situations.

You might not ​fully understand​ what exactly is happening – because the behavior of matter at this scale is ​so​ weird it almost defies explanation – but the tools scientists have developed to describe quantum theory are indispensable to any physicist.

Quantum mechanics is the branch of physics that deals with extremely small particles and other objects on similar scales such as atoms. The term "quantum" comes from "quantus," which means "how great," but in context, it refers to the fact that energy and other quantities like angular momentum take on discrete, quantized values at the scales of quantum mechanics.

This is opposed to having a "continuous" range of possible values, like quantities at the macro scale. For example, in classical mechanics, any value for the total energy of say, a ball in motion, is allowed, while in quantum mechanics, particles like electrons can only take specific, ​fixed​ values of energy when bound to an atom.

There are many other differences between quantum mechanical systems and the world of classical mechanics. For example, in quantum mechanics, observable properties don't have a definitive value ​before you measure them​; they exist as a superposition of multiple possible values.

If you measure the momentum of a ball, you're measuring the real-world, pre-existing value of a physical property, but if you measure the momentum of a particle, you're picking out one of a selection of possible states ​by the act of taking a measurement​. The outcomes of measurements in quantum mechanics depend on probabilities, and so scientists can't make definitive statements about the outcome of any one specific statement in the same way as in classical mechanics.

As a simple example, particles don't have well-defined positions, but have a set (and well-defined) range of positions across space, and you can write the probability density across the range of possible locations. You can measure a particle's position and get a distinct value, but if you performed the measurement again in the ​exact same circumstances​, you would get a different result.

There are many other unusual properties of particles too, such as wave-particle duality, where each matter particle has an associated de Broglie wave. All small particles exhibit both particle-like and wave-like behavior depending on the circumstances.

Wave-particle duality is one of the key concepts in quantum physics, and that's why each particle is represented by a wave function. This is usually given the Greek letter ​Ψ​ (psi) and is a function of position (​x​) and time (​t​), and it contains all of the information that can be known about the particle.

Think about that point again – despite the probabilistic nature of matter at the quantum scale, the wave function allows for a ​complete​ description of the particle, or at least as complete a description as is possible. The output may be a probability distribution, but it still manages to be complete in its description.

The modulus (i.e. absolute value) of this function squared tells you the probability you'll find the particle being described at position ​x​ (or within a small range d​x​, to be precise) at time ​t​. Wave functions have to be normalized (set so that the probability is 1 that it will be found ​somewhere​) for this to be the case, but this is almost always done, and if it isn't, you can normalize the wave function yourself by summing the modulus squared over all values of ​x​, setting it to equal 1 and defining a normalization constant accordingly.

You can use the wave function to calculate the expectation value for the position of a particle at time ​t​, which is essentially the average value you would obtain for the position over many measurements.

You calculate the expectation value by surrounding the "operator" for the observable (e.g. for position, this is just ​x​) with the wave function and its complex conjugate (like a sandwich) and then integrating over all of space. You can use this same approach with different operators to calculate expectation values for energy, momentum and other observables.

The Schrodinger equation is the most important equation in quantum mechanics, and it describes the evolution of wave function with time, and allows you to determine the value of it. It's closely related to the conservation of energy and is ultimately derived from it, but it plays a role similar to that played by Newton's laws in classical mechanics. The simplest way to write the equation is:

\(H Ψ = iℏ \frac{\partial Ψ}{\partial t}\)

Here, ​H​ is the Hamiltonian operator, which has a longer full form:

\(H = -\frac{ℏ^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\)

This acts on the wave function to describe it's evolution in space and time, and in the time-independent version of the Schrodinger equation, it can be considered the energy operator for the quantum system. Quantum mechanical wave functions are solutions to the Schrodinger equation.

The Heisenberg uncertainty principle is one of the most famous principles of quantum mechanics, and states that the position ​x​ and momentum ​p​ of a particle cannot both be known with certainty, or more specifically, to an arbitrary degree of precision.

There is a ​fundamental​ limit to the level of accuracy with which you can measure both of these quantities simultaneously. The result comes from the particle wave duality of quantum mechanical objects, and specifically the way they are described as a wave packet of multiple component waves.

While the position and momentum uncertainty principle is the most well-known, there is also the energy-time uncertainty principle (which says the same thing about energy and time) but also the generalized uncertainty principle.

In short, this states that two quantities which do not "commute" with each other (where ​AB – BA ≠ 0​) can't be known simultaneously to arbitrary precision. There are many other quantities which do not commute with each other, and so many pairs of observables that can't be precisely determined at the same time – precision in one measurement means a huge amount of uncertainty in the other.

This is one of the main things about quantum mechanics that's hard to understand from our macroscopic perspective. Objects you encounter on a day-to-day basis ​all​ have clearly defined values for things like their position and their momentum at all times, and measuring the corresponding values in classical physics is only limited by the precision of your measuring equipment.

In quantum mechanics, though, ​nature itself​ sets a limit to the precision you can measure two non-commuting observables to. It's tempting to think this is simply a practical problem and you'll be able to achieve it one day, but that simply isn't the case: It's impossible.

The weirdness implied by the mathematical formalism of quantum mechanics gave physicists a lot to think about: What was the physical interpretation of the wave function, for example? Was an electron ​really​ a particle or a wave, or could it really be both? The Copenhagen interpretation is the most well-known attempt to answer questions like this and still the most widely-accepted one.

The interpretation essentially says that the wave function and the Schrodinger equation are a complete description of the wave or particle, and any information that cannot be derived from them simply doesn't exist.

For example, the wave function spreads across space, and this means that the particle itself doesn't have a fixed location until you measure it, at which point the wave function "collapses," and you obtain a definite value. In this view, the wave-particle duality of quantum mechanics doesn't mean that a particle is ​both​ a wave and a particle; it simply means that a particle like an electron will behave as a wave in some circumstances and as a particle in others.

Niels Bohr, the biggest proponent of the Copenhagen interpretation, would reportedly criticize questions like, "Is the electron actually a particle, or is it a wave?"

He said they were meaningless, because in order to find out you have to conduct a measurement, and the form of the measurement (i.e. what they were designed to detect) would determine the result you obtained. In addition, all measurements are fundamentally probabilistic, and this probability is built into nature rather than being due to a lack of knowledge or precision on the part of the scientists.

There is still a lot of disagreement about the interpretation of quantum mechanics, though, and there are alternative interpretations that are worth learning about too, in particular the many worlds interpretation and the de Broglie-Bohm interpretation.

The many worlds interpretation was proposed by Hugh Everett III, and essentially removes the need for the collapse of the wave function entirely, but in doing so proposes multiple parallel "worlds" (which has a slippery definition in the theory) coexisting with your own.

In essence, it says that when you make a measurement of a quantum system, the result you obtain doesn't involve the wave function collapsing onto one particular value for the observable, but multiple worlds untangling and you finding yourself in one and not the others. In your world, for example, the particle is at position A rather than B or C, but in another world it will be at B, and in yet another it will be at C.

This is in essence a deterministic (rather than a probabilistic theory), but it's your uncertainty about which world you inhabit that creates the apparently probabilistic nature of quantum mechanics. The probability really relates to whether you're in world A, B or C, not where the particle is within your world. However, the "splitting" of worlds arguably raises as many questions as it answers, and so the idea is still quite a controversial one.

The de Broglie-Bohm interpretation is sometimes called ​pilot wave mechanics​, and it follows from the Copenhagen interpretation in that particles are described by wave functions and the Schrodinger equation.

However, it states that every particle has a definite position even when it isn't being observed, but it is guided by a "pilot wave," for which there is another equation you use to calculate the evolution of the system. This describes the wave-particle duality by essentially saying that a particle "surfs" at a definite position on a wave, with the wave guiding it's motion, but it still exists even when not observed.