Johannes Loevenich
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Quantum Mechanics: The Wave Function

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Quantum Mechanics: The Wave Function

How It Works, And Why Understanding This Is Important!

Johannes Loevenich
·Dec 6, 2021·

8 min read

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The quantum world is a strange and puzzling place. It's so different from our everyday experience that it can be difficult to wrap our heads around it. One of the biggest mysteries of quantum mechanics is the wavefunction, which describes what particles are doing before they're measured. There are many interpretations for how this wave function works, but one thing remains true: quantum mechanics has an indeterminacy about its behavior that we just don't have in the classical world. We will explore some of these interpretations and see how quantum theory gives rise to electron interference!

The importance of understanding this for future scientists

One of the most important things we learn from quantum mechanics is that there is inherent uncertainty in nature. This uncertainty isn't something we can ignore or sweep under the rug- it's a fundamental part of how the world works on a quantum level. It's this uncertainty that leads to strange and unpredictable behavior, like electron interference. Future scientists need to understand quantum mechanics in order to make sense of these puzzling phenomena. Even if they don't end up working in quantum physics themselves, the understanding quantum theory will help them think more deeply about the natural world and all its mysteries!

A brief history on how we got here - from Newtonian physics to quantum mechanics

Isaac Newton was able to describe the motion of objects in a very accurate way. He could combine his laws of motion with Galileo Galilei's observations about falling bodies, and came up with some fundamental rules that govern how things move under gravity. As our technology improved over time, scientists saw quantum phenomena occurring on scales they couldn't explain using classical physics- namely black body radiation! Max Planck decided that if he wanted to understand quantum behavior, it would be necessary to build a new theory from scratch that included discrete units for energy (quanta). Albert Einstein took this idea further by suggesting light behaved as both particles AND waves. Niels Bohr used these ideas along with Louis de Broglie's hypothesis about matter waves to come up with quantum mechanics.

What is the Wave Function in Quantum Mechanics

Put simply, the wavefunction is a mathematical representation of all the possible states that a quantum system can be in. It's basically like a blueprint for all the potential configurations that a particle could be in. This includes not just where it is, but its momentum and energy as well.

How does it work?

When you measure a quantum system, its wavefunction collapses down to one specific state. The reason we see quantum mechanics as so indeterminate is that there are multiple possibilities for how the wave function might collapse when we make a measurement. It's sort of like flipping a coin- you have two options, heads or tails, and until you actually flip the coin, both are equally likely outcomes.

How do you use the wave function to calculate things like position, momentum, and energy?

The wave function gives you all the information that quantum mechanics requires to tell us how a particle might behave and it's similar to having detailed instructions for building something- it tells us where every piece goes, and what shape it should be (though exactly HOW the pieces come together is up to our own interpretation). We will learn about the Schroedinger equation describing the wave function in the next paragraph.

The Schroedinger Equation

The wavefunction in quantum mechanics is described by the Schroedinger equation. This equation was developed by Erwin Schroedinger and is one of the most important equations in quantum mechanics. It describes how the wave function changes over time and helps us to understand the behavior of particles on a quantum level.

To get an understanding of Schoedinger's Equation, let's start with vectors! A vector has both direction and magnitude. The first two letters in "vector" give us those two pieces of information: v for direction and t for magnitude. This is not quantum mechanics, after all, so let's translate that into quantum mechanical notation!

A vector in the quantum world looks like this: |ψ⟩. The little hats on top indicate that we're talking about a quantum state instead of just any old state. The quantum state is a vector that has direction and magnitude, just like the vectors we're used to in classical mechanics!

Schroedinger's Equation tells us how |ψ⟩ changes with time. But what does this all mean? We can look at it from three different perspectives: either as an operator, as a wave, or as a statistical tool.

The Wave Interpretation

When we think of quantum mechanics as a wave, the equation tells us how the wave changes over time. This is what Schroedinger was thinking about when he came up with his equation, he wanted to find a way to describe the changing waves in quantum physics.

This interpretation is helpful for understanding some of the quantum mechanical mysteries, like electron interference. But it's important to note that this wave is not a physical object- it's just a way of describing how the quantum state changes over time.

The Operator Interpretation

The quantum state can also be understood as an operator. Operators are function-like things that act on quantum states to produce another quantum state, just like how functions work in the classical world! When we understand this wavefunction is acting on quantum states according to Schroedinger's equation, it makes a lot more sense why it looks so strange.

One of the most famous consequences of the Schroedinger equation is Heisenberg's Uncertainty Principle, which states that certain properties of particles (like momentum) cannot be known with absolute certainty. This principle is inherent in quantum mechanics and arises from our inability to measure all aspects of a particle at once.

Born's Statistical Interpretation

In 1926, Max Born suggested that the wavefunction was a statistical tool. In quantum mechanics, you can't predict which path an electron will take between two points in space—it's only possible to calculate probabilities of where it might be at different times. This is known as quantum indeterminacy and means we can never know everything about a particle at once. If this sounds strange, consider how difficult it would be for us to measure momentum or position if there were no way to guess anything about their future behavior! Since quantum mechanics has inherent uncertainty built into its equations, the best thing we have going for us is probability distributions when trying to understand what particles are doing before they're measured.

But, Born had another mind-blowing idea: he suggested that the wavefunction is not just a description of particles, but actually creates them. This is called quantum ontology and it's one of the more controversial interpretations of quantum mechanics.

The collapse of the Wavefunction: You talk about measurement, but where was the particle just before the measurement?

It's impossible to say because there are so many possibilities! You can ask yourself instead: what were the probabilities for finding my quantum system in each of those different configurations? That will tell you how likely you are to find your quantum object at every point in space, which is good enough for most purposes.

When we talk about the collapse of the wavefunction, we're talking about what happens when a quantum system is measured. Remember that in quantum mechanics, you can't predict which path an electron will take between two points in space—it's only possible to calculate probabilities of where it might be at different times.

But, when you measure a quantum system, all of its possibilities collapse down into just one outcome. This is called measurement resolution and it's why quantum mechanics is so powerful- by measuring particles we can determine their exact state, something that's not possible classically!

Electron Interference Example:

One great way to understand quantum mechanics is through examples! Let's take electron interference as an example. In quantum mechanics, it's possible for particles like electrons and photons to interfere with each other! This means that they can come together and their quantum waves will combine: these quantum objects are said to be coherent. We even see this kind of interference in everyday life- if you take two metal plates and put them close enough together, light coming from one side will bounce off both plates at once before heading out into space on the other side. The result? You'll get some regions that look very bright (where lots of light interfered constructively), and some darker regions where there wasn't as much interfering going on (where lots of light interfered destructively). This is quantum interference (constructive and destructive interference) at work!

The Three Main Interpretations

There are three main interpretations of where particles were just before they're measured: realistic, orthodox, and agnostic.

Realistic interpretation: The wave function exists independently of any observer or measuring device. It describes a physical reality that exists beyond our ability to measure it.

Orthodox interpretation: The wave function only exists when it's being observed. It's a tool used to predict probabilities but doesn't have any real existence outside of this context.

Agnostic interpretation: There is no right answer to this question! Some people believe quantum mechanics is real while others think it's just a math game.

It's up to the individual to decide what they believe.

Which interpretation you choose really depends on your personal beliefs about quantum mechanics. No one interpretation is right or wrong- it's all a matter of perspective!

Conclusion

Quantum mechanics is a very complex topic and there are many different interpretations of what it means. In this blog post, we've talked about the wave function in quantum mechanics- what it is and how it works. We've also looked at Schroedinger's Equation and the different interpretations of it, as well as the three main positions on where particles were just before they're measured. I hope this gives you a better understanding of quantum mechanics and its complexities! Thanks for reading!

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