Quantum Superposition and Wave Functions
A clear, beginner-friendly guide to superposition, wave functions, and the math behind quantum weirdness.
Introduction to Quantum Mechanics
Quantum physics is a complex form of physics that focuses on the behavior of particles at extremely small scales, where the rules of classical physics no longer apply. Instead of having definitive states and positions, quantum systems are explained using wave functions, which are mathematical descriptions of the possibilities of a particle or a system’s position, momentum and other factors. Two famous examples of quantum behavior are Schrödinger’s cat and the double‑slit experiment, both exploring the use of wave functions and superposition. Though quantum mechanics can feel like strange speculation, it relies heavily on mathematics to analyze and predict systems. The aim of this paper is to explore how quantum superposition and wave interference are shown through a wave function equation. It will also show how the mathematical structure is used in the case of Schrödinger’s cat and the double slit experiment. Further, the uses of the quantum mathematics and the purpose of quantum mechanics in the world will be shown, making it clear that although you may never have heard of it before, it has an important role in our lives.
Schrödinger’s Cat Thought Experiment
In 1935, Austrian scientist Erwin Schrödinger asked the question: where does quantum physics end and classical physics begin? Quantum physics describes the behavior of extremely small particles such as electrons or radioactive atoms, which by the laws of quantum mechanics, may exist in multiple states at once. Schrödinger used a wave function to analyze the possibilities of decay of a radioactive particle; the outcomes would be either decay or not decay.
However, wave functions are only used for extremely small particles. Schrödinger wanted to see where the boundary is between quantum and classical physics. He wondered, knowing that a particle can exist in multiple states at once (until analyzed), what would happen when that particle is tied with something on a larger scale, like a cat.
Schrödinger’s thought experiment was designed around showing the connection between quantum and classical rules. His experiment, which was purely theoretical, was to put a radioactive particle in a box with a cat, if the particle decays to half its original mass, then a vial of toxic gas would be released and kill the cat. The radioactive particle, by quantum rules, is both decayed and not decayed at the same time until observed. Schrödinger wondered whether the cat, whose fate depended on a particle in two states, should also be considered both alive and dead? Of course, the cat isn’t actually both dead and alive at the same time, but Schrödinger wanted to show what quantum mechanics would look like if used on macroscopic systems.
The Double Slit Experiment
The second experiment to demonstrate quantum principles is called the double slit experiment. This experiment is very strange, as it demonstrates that particles like electrons and protons can interfere as waves, even when sent one at a time. To break it down, we often imagine particles as tiny balls moving in straight lines; however in quantum conceptualization, they treat the particles as waves (similar to ripples in water). The experiment is set up as follows; a wall is set up with two slits parallel to one another, additionally a light sensitive screen is placed behind the wall. A single particle is sent out and acts as if it is a wave, the single particle goes through both slits at the same time and when they go through the wall, two new waves originate from the slits from the initial one. From that point, the waves interfere with one another, and although being the same particle, they create what is called an interference pattern that is shown on the light sensitive screen. The pattern is a series of dark and light stripes lined up to one another. This result shocked the scientists who performed the experiment, as the classical physics they applied to the particle did not predict this behavior. The classical physics predicted two lines on the screen matching up with the slits the particles were shot through; however, the classical physics did not predict an interference pattern, showing that on such an incredibly small scale, a new form of physics must apply. This realization gave birth to the foundations of quantum physics.
However, the experiment shifts our conception in that when sensors are put near slits to determine which slit the particle will go through, and where it will land on the screen. When the sensors are placed, the particle still behaves as a wave, but when it nears the detector, the act of measuring forces the particle to stop acting as a wave (and going through both slits) and to behave as a particle (and go through one slit or the other). This means that when the particles are unobserved, they create the interference pattern, but when observed, behave as classical physics says they will and create corresponding lines on the screen, twisting what we had thought that we once knew. Anne Helmenstine at Sciencenotes.org notes:
These modern versions of the experiment confirmed that light (and later, electrons, atoms, and molecules) exhibits both wave-like and particle-like properties, depending on how we observe it. This duality became one of the central concepts in quantum mechanics, and the double slit experiment evolved from a classical test of light’s nature into a profound exploration of quantum reality.
This experiment, like Schrödinger’s cat, shows a superposition, in this case when it is not observed, where the particle goes into both slits at once, just as Schrödinger’s cat is supposed to be in two states at once. When the Cat or the particle is observed the dual possibilities collapse into one, either the cat is dead or alive, and the waves revert into behaving as particles. Both experiments rely on the same mathematical concept: a wave function that contains multiple possible states until observation forces one outcome.
Understanding Superposition
Superpositions, which describe a state of being, defy the laws of our typical understanding. To understand this, in classic logic a coin will either be heads or tails, but quantum logic says that before you look at the coin, it is described by a state that includes both possibilities, both heads and tails. Not physically both heads and tails, but mathematically both.
Wave Functions and Equations
In quantum mechanics, the method scientists use to demonstrate the outcomes of a quantum system is to express them in a wave function which is the sum of both possibilities. Wave functions can be applied to show the fate of Schrödinger’s cat before observation, as well as which slits the particle may go through, with slight modifications to each to make them properly assess the situation. The base equation looks like this:
ψtotal = ψ₁+ ψ₂
This equation is only used in both cases as the base for a more complex equation, both scenarios having a specialized form. The two equations, Schrödinger’s cat and the double slit equations look like this respectively:
ψcat = a|alive⟩ + b|dead⟩
P= |ψ₁|²+|ψ₂|² + 2Re(ψ₁ψ₂*)
Application to Schrödinger’s Experiment
Quantum mechanics has a rule that states “the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared” (Wikipedia, n.d.). Simply put, this means that when there is superposition of possibilities, the probability of the outcome is given by the square of the magnitude of its amplitude. For example, in the Schrödinger’s cat experiment, with the variables of dead or alive, the two are given the amplitudes a and b, so:
P(alive)= |a|² and P(dead)=|b|²
These equalities can be used to solve the probability of Schrödinger’s cat fate. If a radioactive atom is used in the experiment, with a 40% chance of decaying in an hour, then there is a 40% chance the cat dies, and 60% chance that it lives. Used in the equation, it looks like this:
P(alive)=0.60 and P(dead)=0.40, meaning that |a|²=0.60 and |b|²=.40.
Meaning the probability amplitudes are:
a=√.60 ≈ 0.7746 and b=√.40 ≈ 0.6324
After gathering the amplitudes, they can be plugged into the original equation in a form that represents the state, telling us exactly how likely each outcome is. Giving us:
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ψcat = 0.7746|alive⟩ + 0.6324|dead⟩
This is the final quantum state expression that displays the information in the form quantum science uses to interpret probabilities.
Application to the Double Slit Experiment
Breaking down the base equation, ψtotal = ψ₁+ψ₂, the terms represent the following:
Ψ₁= The wave function for the particle going through slit one
Ψ₂=The wave function for the particle going through slit two
This is a superposition, just like the cat equation, except now the states are: “went through slit one” and “went through slit two”. Both use the same math but have their solutions convey different meanings. Using the same rule as before, the equation becomes P=|ψ₁+ψ₂|². At this point, we can expand the quantum equation to a more fitting expression, and it becomes:
P=|ψ₁|²+|ψ₂|²+2Re(ψ₁ψ₂*)
This is an expanded version of the base, called the general interference formula, it adds 2Re(ψ₁ψ₂*) to the base, which is the interference term. This formula shows where the interference pattern (as explained above) becomes mathematically represented. However, this is not the final equation, there is one last segment added to the equation above, used in the simplest possible wave functions: ψ₁=A and ψ₂=Aeiϕ. A represents the amplitude, eiϕ adds what is called a phase difference between the two waves, with ϕ determining whether the waves reinforce or cancel. This probability equation evolves into its final version, becoming P=2A²(1+cos ϕ). The final step to determine the interference pattern on the wall is plugging in the values into the equation. Since cos(ϕ) changes between 1 and -1, the probability creates alternating bright and dark bands at the different stages. This is evident when ϕ is equal to 0, we get P=4A2, telling us that the interference creates a band of maximum brightness. When ϕ is equal to π, then P=0, telling us that there is now a band of complete darkness. This is exactly what the screen reveals in the experiments illustrated below:
Figure 1. Double slit experiment setup showing electrons creating an interference pattern.
This image illustrates the process and shows the result of the pattern the interference, proving the equation properly represents the outcome of the experiment.
Practical Use of This Quantum Knowledge
Although these experiments seem strange, the concepts and equations related to them have a more practical use, capable of being applied as a tool throughout the use of quantum physics.
Uses of Schrödinger’s Equation
Apart from being used to determine if a cat is alive or not, Schrödinger’s equation is often used to predict scenarios such as the chance an electron is at a certain energy level, the chance a photon is detected in a certain place, and the chance a qubit is in a state of 1 or 0.
To demonstrate its use, we can use the equation to predict the state of a qubit in a certain state. According to Wikipedia (2026), a qubit is the quantum version of a classical computer bit, capable of being 0, 1, or both at the same time through superposition. Supposing, hypothetically, that a qubit is in the state: ψ=(√3/2) |0⟩ + (1/2) |1⟩, then when separately solved:
The probability of 0 =( √3/2)² = (3/4) = 75%
And
The probability of 1= (1/2)² = (1/4) = 25%
Showing that the probability of the qubit being 0 is a 75% chance, and the probability of it being 1 is a 25% chance. Situations like this are an example of real-world applications of Schrödinger’s equation, allowing scientists to predict outcomes of similar systems.
Uses of Wave Function Equations
The equation P=|ψ₁ψ₂|², which I modified for the double slit experiment, can also be used for fields using electron microscopes, microengineering, and MRIs. In an MRI, radio waves are used to interact with the polarity of the hydrogen atoms in the body (mostly from water), “Creating a uniform magnetic signal throughout your body” (ScienceInsights, 2026). The hydrogen atoms, before the machines’ magnetic field is applied, are considered to be in a superposition, because their magnetic directions aren’t fixed, similar to the particles in a wave function.
When the radio waves are used during the MRI, the hydrogen atoms are briefly disturbed and return to their original positions. This disturbance emits signals that can be added together, creating a constructive interference, (similar to when the waves reinforced, creating a light band in the double slit experiment), or cancel out, creating a destructive interference, (like when the waves canceled each other to make a dark band). The same modified equation, P=|ψ₁|²+|ψ₂|²+2Re(ψ₁ψ₂*), can be applied to describe how these signals function.
For example, suppose there are two hydrogen atoms both with different “wave strengths” of a signal, ψ₁= 0.5, and ψ₂= 0.7, assuming a small phase difference. We can apply the equation:
P= |ψ₁|²+|ψ₂|²+ 2Re(ψ₁ψ₂*)
P= 0.25 + 0.49 + 0.7(0.707)≈1.24.
What this number tells us is how strong the signal is from that specific location. The calculation process is repeatedly used across many atoms, allowing the MRI to create detailed images of soft tissues inside the body.
Conclusion
Although an obscure field, quantum physics is very important for providing a greater understanding of how particles behave in situations where classical physics is unable to explain. Schrödinger’s thought experiment demonstrates the function of a superposition, where a system (in his case a cat) can exist in multiple states at once until observed. The double slit experiment furthers the understanding of quantum mechanics, demonstrating the use of wave functions patterns, and showing how observation affects outcomes. Both experiments exemplify different uses of quantum mechanical math, showing how both their outcomes are backed by the calculations.
These experiments may be abstract and strange, but they have real-world applications: Schrödinger’s equation can be used to predict systems like qubit states in quantum computing, and wave interference principles can be applied to MRI technology, providing greater insight into everyday implementations.
Quantum mechanics challenges our conception of reality, suggesting ideas that seem unusual but are supported by the math, revealing that reality is much more complex than we may have thought. Overall, quantum mechanics gives us a deeper understanding of the world we cannot see, yet has great importance in our lives.
