What is The Schrödinger Equation, Exactly?

Summary

In this educational video, Jade demystifies the complex Schrodinger equation, a fundamental concept in quantum mechanics. She likens it to the 'F=ma' of the quantum world, explaining how it reveals everything we can know about a quantum system. Jade breaks down the equation's components, including the wave function (denoted by the Greek letter psi), which describes the probability distribution of an electron's location, and the allowed energy levels. She also touches on the quantization of energy levels, a key feature of quantum mechanics, and the difference between potential and kinetic energy in this context. The video aims to provide viewers with a clearer understanding of the Schrodinger equation, enabling them to grasp quantum concepts better.

Takeaways

• 🧲 The Schrodinger equation is fundamental in quantum mechanics, often compared to F=ma in classical physics.
• 🌌 It encapsulates everything we can know about a quantum system, including energy levels and wave functions.
• 🎲 The wave function (represented by the Greek letter psi) provides probabilities of where a quantum object, like an electron, is likely to be found.
• 🚫 The Heisenberg Uncertainty Principle limits our ability to know both the exact position and momentum of quantum objects simultaneously.
• 🔮 The wave function describes a quantum object's probability distribution, which is spread out like a wave, unlike classical objects that occupy a single location.
• 📉 Energy in quantum systems is quantized, meaning particles can only have certain discrete energy levels, unlike the continuous range in classical physics.
• ⚖️ The Schrodinger equation balances a quantum system's total energy as the sum of its kinetic and potential energies.
• 🔑 Solving the equation yields the energy levels and wave functions, which are key to understanding a quantum system's behavior.
• 🎓 The solutions to the Schrodinger equation involve constants and whole numbers, indicating that energy and probability distributions are quantized.
• 🔗 The script suggests using resources like Brilliant.org for a deeper understanding of quantum mechanics and the Schrodinger equation.

Q & A

What is the significance of the Schrödinger equation in quantum science?

The Schrödinger equation is fundamental in quantum science as it encapsulates everything we can know about a quantum system, much like F=ma in classical physics. It allows us to predict the energy levels and wave functions of quantum objects.

Why doesn't the classical equation F=ma apply to quantum particles?

The classical equation F=ma does not apply to quantum particles because at the quantum level, the Heisenberg uncertainty principle prevents us from knowing both the exact position and momentum of particles simultaneously, necessitating a different approach like the Schrödinger equation.

What is the wave function (Ψ) in the context of the Schrödinger equation?

The wave function (Ψ) represents the probability distribution of a quantum particle's location. It tells us where a particle is likely to be found but does not pinpoint an exact position until measured.

How does the Heisenberg uncertainty principle relate to the Schrödinger equation?

The Heisenberg uncertainty principle, which states that the exact position and momentum of a quantum object cannot be simultaneously known, is inherent in the Schrödinger equation through its wave function, which only provides probabilities of location.

What does the energy term (E) in the Schrödinger equation signify?

The energy term (E) in the Schrödinger equation signifies the allowed energy levels of a quantum system. These levels are quantized, meaning the particle can only have certain discrete energy values, reflecting the wave-like nature of quantum objects.

Why are energy levels quantized in quantum mechanics?

Energy levels are quantized in quantum mechanics due to the wave-like nature of particles. The wave function must be zero at the boundaries (like the walls of a box), which restricts the allowable frequencies and, by Einstein's relation E=hf, the allowable energy levels.

What is the role of potential energy (V) in the Schrödinger equation?

In the Schrödinger equation, potential energy (V) is a term that, along with kinetic energy, contributes to the total energy of the system. It represents the energy of the particle due to its position in a potential field.

How does the Schrödinger equation help in determining the behavior of an electron in a box?

The Schrödinger equation helps determine the behavior of an electron in a box by providing solutions for the energy levels and wave functions. These solutions predict the probabilities of where the electron is likely to be and the allowed energy states it can occupy.

What is the significance of the quantized values in the solutions to the Schrödinger equation?

The quantized values in the solutions to the Schrödinger equation are significant because they represent the discrete states that a quantum system can exist in. These values are whole numbers and constants, indicating that the system's properties are not continuous but occur in specific, quantized states.

How does the act of measurement affect the wave function of a quantum particle?

The act of measurement causes the wave function of a quantum particle to collapse from a superposition of probabilities to a single outcome. This means the particle transitions from being in all possible states at once to being in a defined state upon observation.

What resources are available for someone looking to delve deeper into quantum mechanics and the Schrödinger equation?

For those interested in further study, platforms like Brilliant.org offer courses on quantum mechanics, starting from foundational experiments to the derivation of the Schrödinger equation, complete with examples and problems to solve.

Outlines

00:00

🧲 Introduction to the Schrodinger Equation

Jade introduces the Schrodinger equation, a fundamental concept in quantum science, which is often mentioned in articles but not always explained. She aims to clarify what it means, comparing it to the F=ma equation in classical physics. The Schrodinger equation is described as encapsulating everything we can know about a quantum system, including its wave function and energy levels. Jade explains that unlike classical physics, where the position and momentum of an object can be determined, quantum mechanics is governed by the Heisenberg uncertainty principle, which limits our knowledge to probabilities and energy levels. The wave function, represented by the Greek letter psi (ψ), provides the probability distribution of where a quantum particle, like an electron, is likely to be found. The video promises to delve into the mathematics behind these concepts, starting with the time-independent version of the equation.

05:02

🔬 Quantum Mechanics and Energy Quantization

This paragraph delves into the quantization of energy levels in quantum systems, using the example of an electron in a box. Jade explains that the energy levels an electron can have are not continuous but discrete, due to the wave-like nature of quantum particles. The wave function must be zero at the boundaries of the box, which restricts the allowed frequencies and, by Einstein's relation E=hf, the allowed energy levels. This quantization is a key feature of quantum mechanics. The paragraph also discusses the composition of energy in quantum systems, which includes both kinetic and potential energy. The potential energy is represented as a function V, and the kinetic energy is derived from the wave function. Jade provides an example of solving the Schrodinger equation for a specific scenario, revealing that the solutions result in quantized energy levels and specific wave functions. The solutions are expressed in terms of constants and whole numbers, emphasizing the quantized nature of quantum states. The video concludes with a reference to resources for further learning, including a course on quantum mechanics and a link for a discount on Brilliant.org.

Keywords

💡Schrodinger Equation

The Schrodinger Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. In the video, it is likened to the F=ma equation in classical physics, providing a framework to predict the behavior of quantum systems. The equation is central to understanding the wave function and energy levels of particles, such as electrons in a box, which is a key concept discussed throughout the script.

💡Wave Function

The wave function, denoted by the Greek letter psi (ψ), is a mathematical description of the quantum state of a particle. It provides information about the probabilities of finding a particle in different locations. In the video, the wave function is used to explain the likelihood of an electron being in a particular place within a box, emphasizing the probabilistic nature of quantum mechanics.

💡Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to know both the exact position and momentum of a quantum object simultaneously. This principle is mentioned in the video to highlight the limitations of quantum mechanics compared to classical mechanics, where precise measurements are possible.

💡Quantum System

A quantum system refers to a system that is governed by the rules of quantum mechanics. In the context of the video, an electron in a box is used as an example of a quantum system. The Schrodinger Equation is used to describe and predict the behavior of such systems, which is a central theme of the video.

💡Energy Levels

Energy levels in quantum mechanics refer to the specific values of energy that a quantum system can have. The video explains that these levels are quantized, meaning they are discrete and not continuous as in classical systems. The Schrodinger Equation is used to determine these allowable energy levels for a system, such as an electron in a box.

💡Probability Distribution

Probability distribution in the context of quantum mechanics describes the likelihood of a particle being found in a particular state or location. The video uses the concept to explain how the wave function collapses to a specific location upon measurement, which is a key aspect of quantum behavior.

💡Superposition

Superposition is a principle in quantum mechanics where a particle can exist in multiple states simultaneously until it is measured. The video references Schrödinger's cat thought experiment to illustrate superposition, where the cat is both dead and alive until observed.

💡Kinetic Energy

Kinetic energy is the energy that a particle possesses due to its motion. In the video, the kinetic energy of a quantum system is discussed in relation to the potential energy, and how together they make up the total energy of the system. The Schrodinger Equation is used to calculate these energies.

💡Potential Energy

Potential energy is the stored energy of a system, which can be converted to kinetic energy. The video explains how potential energy is represented in the Schrodinger Equation and how it contributes to the total energy of a quantum system.

💡Quantization

Quantization refers to the property of a physical quantity to be discrete, rather than continuous. The video discusses how energy levels in quantum systems are quantized, meaning they can only take on certain specific values, which is a fundamental concept in quantum mechanics.

💡Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. In the video, it is mentioned in the context of the equation E=hf, which connects energy and frequency, and is crucial for understanding the quantization of energy levels.

Highlights

The Schrodinger equation is central to quantum mechanics, providing a framework to understand quantum systems.

It is likened to F=ma in classical physics, offering insights into the quantum world's dynamics.

The equation encapsulates all knowable properties of a quantum system, including energy levels and wave functions.

The wave function, denoted by the Greek letter psi (ψ), predicts the probability distribution of a quantum particle's location.

Quantum objects like electrons exist in a superposition of states until measured, unlike classical objects.

The Heisenberg uncertainty principle limits the precision with which both position and momentum can be known simultaneously.

The energy levels of a quantum system are quantized, differing from the continuous values in classical physics.

The Schrodinger equation's solutions reveal allowed energy levels and wave functions, essential for predicting particle behavior.

The potential and kinetic energy within the equation are crucial for determining a system's total energy.

The equation's time-independent version is discussed, focusing on stationary states without explicit time dependence.

Electrons in a box model is used to illustrate the application of the Schrodinger equation to confined quantum systems.

The wave function's nodes, where it equals zero, are critical for determining allowed energy levels.

Planck's constant links energy and frequency, showing that only certain frequencies and thus energy levels are permissible in a quantum system.

The solutions to the Schrodinger equation for an electron in a box are quantized, reflecting the discrete nature of quantum mechanics.

The video provides a simplified explanation of the complex mathematics behind the Schrodinger equation.

The probability distribution of an electron's location is visualized through the wave function's squared magnitude.

The video encourages viewers to explore quantum mechanics further through educational resources like Brilliant.org.

The presenter shares personal experiences and insights from studying quantum physics, emphasizing the importance of building intuition.

A discount link to Brilliant.org is provided for those interested in a comprehensive course on quantum mechanics.

The video concludes with an invitation for feedback to improve future educational content on quantum physics.

Transcripts

Hi guys! I'm Jade, lovely to meet you. So the Schrodinger equation is one of those

things that pops up a lot in like quantum science articles and journals

and stuff but the journalist doesn't usually go into what it means, which is

fair because it's a fairly complex topic. So today I just wanted to share with you

guys like what it actually means so next time you read it in an article you can

get a better gist of what it's about. So the short version is the Schrodinger

equation tells us everything we can possibly know about a quantum system.

It's basically the F=ma of the quantum world. If you throw a ball and

solve F=ma you can predict its position and

momentum for any moment in time. Once you have these two you can derive basically

everything else you could possibly know about it, velocity, energy etc. But when we

get down to particle land Newton's equations don't work anymore.

If we put a particle in a box and we want to know where it is F=ma just

doesn't cut it. The Heisenberg uncertainty principle says that we can't

know both the exact position and momentum of quantum objects, but, we can

know other things, like the energy levels and the wave function, which we'll explore

in this video. But that information is all inside the Schrodinger equation and

with some hardcore mathematics we can tease it out. Now the long version! So

before I get started I should say that this is the time independent version

meaning that it doesn't involve time. And if at any point throughout the video you

get lost, don't worry, even Schrodinger didn't know exactly what his equation

meant. So let's say we have a quantum system: an electron in a box. We want to

know everything we can about this electron so we can make predictions,

where it might be, what energy level it might be at. These

answers are all buried within the Schrodinger equation. So first let's

start with this guy, the pitchfork thing. This is the Greek letter psi and it

stands for what's called the wave function. It tells you where the electron

is likely to be. But not where it will be. See quantum objects are sneaky in that

you can't predict exactly where they'll be until you measure them. You can only

predict where they'll probably be. Say there's this kid you know. You've grown

up with them your whole life. They're locked in their room with their homework, a

Playstation, and a bed. If you had to guess where they were you'd say there's

about an 80% probability they're on the PlayStation, 19% probability they're

in bed and a 1% probability they're doing their homework. When you open the

door you'll know for sure. But you were able to make these predictions because

you know this guy. What if you had to guess where this electron is? You don't

know this electron. Well that's what the wave function tells us. It gives us the

probabilities of where it's likely to be. But a big difference is that while the

guy is only in one place at a time the electron is in a superposition of all

possible places at the same time. You may have heard the famous thought experiment

Schrodinger's cat who's in a superposition of being dead and alive

until the box is opened and it's forced to choose a state. It's the same deal

here. The act of not knowing where the electron is allows it's probability

distribution to be spread out over a large space kind of like a wave.

Different kinds of waves can represent different probabilities of where it's

likely to be, hence the name wave function. It's a function that describes

the wave shape of probability distribution of the electron.

Oh yeah and when you open the door and measure where it is this wave

probability cloud function collapses and the electron becomes a particle again. No

wonder Schrodinger was confused. So that's what the pitchfork means, the

wave function tells us where our electron is likely to be. Now let's take

a look at this E. It represents the energies the electron is allowed to have.

Now before I go into what that means I just want to point out that the way this

equation is arranged these values are the ones we're trying to solve for, so

it's telling us that hey if you do all this stuff you can find out the energy

levels of the wave functions of the electron! And if we know these two things

we can derive everything else we can possibly know about the particle, just

like the position and momentum of the ball. But let's backtrack a sec, what do I mean

when I say energy levels the electron is allowed to have? Like it's a grown

electron it can have whatever energy levels it wants right? Well, no. In the

regular world we see around is, energy can go up and down in a

smooth continuous way, but this isn't the case in the quantum world and the reason

comes from the wave-like nature of the probability distributions. Because our

particle is inside the box it has a zero probability of being found in or outside

the wall, so this means the wave function always needs to be zero there otherwise

there's some probability that the electron could be outside the box which

we know it's not. That means that the electron can only have certain

frequencies associated with it. This frequency is allowed as the wave

function is zero at both edges, and this frequency is not, this frequency is

allowed and this one is not. So Einstein discovered that energy is actually

proportional to frequency by this relation E=hf where E is the

energy, f is the frequency and this h here is Planck's constant. Don't worry

too much about that h for now, all you need to know is that it's a constant

which means its value doesn't change. So if only certain frequencies are allowed

inside the box and this is a constant then it follows that only certain energy

levels are allowed inside the box too. This property of discrete or quantized values

is where quantum mechanics gets its name. Things that can take on continuous

values in the regular world like energy levels can only take on certain

quantized values at the quantum scale. Now let's look at the other side of the

equation. We know what we're solving for: the energy levels and the wave functions. But how is all of this going to help us get there? Well overall energy is made up

of kinetic energy and potential energy. If a skateboarder is on a skate ramp

they'll be traveling at some speed and have some kinetic energy, but when they

stop at the top they still have energy, it's just transformed into a different

kind: potential energy. The entire energy of the system is just the kinetic energy

plus the potential energy and while it can move back and forth between those

two states its total value is always conserved. Sometimes potential energy is

written as a V, so this term is the potential energy of the wave function. So

if this is the potential energy that must mean that this term here is the

kinetic energy I know it doesn't really look like any kind of kinetic energy

equation we've seen before right, so here's the derivation if you don't

believe me. So if we can solve for the potential and

kinetic energy of our quantum system this will tell us the energy levels

allowed, and that's everything there is to know about our little electron! So

what would some typical Schrodinger solutions look like? Well to this

particular problem all the solutions to the wave functions

take these two forms and the energy equation that popped out was this. Great

Jade so what the hell does that mean? Well the first thing to note is that

every term in this expression is either a constant or a whole number: h-bar is a

constant, 2 is obviously a constant, m the mass of the electron is a constant, pi is

a constant, and L the length of the box is a constant, and n stands for the

different states of the electron and they're all whole numbers, 1, 2, 3, etc. So

then the energy E can only have certain values. It's quantized. But what about the

wave function? Where is the electron? Well let's look at this guy when the electron

is in its first energy state, when n is equal to 1. We get this. That's one of the

wave functions of the electron and if we square it we get the probability

distribution, aka where the electron is likely to be.

We can see that there's a high probability that it'll be found in the

middle here but a zero probability it'll be found right at the edges. Here

are some more wave functions and probability densities for other energy

states. See how the wave function is always always 0 right at the edges. This

took me an entire semester of a physics degree to understand and what really

helped me was working through a lot of problems and taking the time to build a

strong intuition. Brilliant.org has an entire course dedicated to quantum

mechanics which starts with the experiments which first discovered

quantum behavior and leads up to the derivation of the Schrodinger equation.

It has examples you can work through at your own pace. I actually just went

through it to refresh my memory and learned some things I didn't know and

definitely understood some things better. They also have heaps of other courses

mainly on physics, math and computer science and they're always adding more.

The first 200 people to click the link below and sign up will get a 20%

discount. Just go to brilliant.org/UpandAtom. The link is on screen and in

the description. And if you're wondering why I didn't include the math behind the

solutions to the Schrodinger equation it's because it would have taken me

about an hour just to write out probably another five weeks to explain,

but for those of you who are especially curious I've posted my final exam essay

on quantum physics in the description. It includes the derivation of the

Schrodinger equation as well as the solutions and some other interesting

stuff like random questions to my professor. So guys please be honest, do

you understand the Schrodinger equation a bit better now? I hope you do but if

you don't please feel free to give me any feedback on how I can make my

explanations better for future videos. Quantum physics is so cool and I really

don't want anybody to miss out. So yeah that's all from me. Bye! -