The Hydrogen Atom, Part 1 of 3: Intro to Quantum Physics

Summary

Dieses Video skizziert die Grundlagen der Quantenmechanik anhand des Wasserstoffatoms, das aus einem Protonen und einem Elektron besteht. Es erklärt, wie das Elektron trotz der Anziehungskraft nicht auf das Proton kollabiert und wie die Quantenunschärfe die Bewegung des Elektrons beeinflusst. Der Fokus liegt auf der Darstellung der Wellenfunktion, der Wahrscheinlichkeitsdichte und dem sogenannten reduzierten Planckschen Wirkung, die die Beziehung zwischen Energie, Frequenz und Impuls definiert. Darüber hinaus wird die Schrödinger-Gleichung vorgestellt, die die Bewegung des Elektrons beschreibt, und es wird die Lösung dieser Gleichung in sphärischen Koordinaten diskutiert.

Takeaways

• 🌌 Der Wasserstoffatom ist ein Portal zwischen der Welt der Experimente und der seltsamen Welt der Quantenmechanik.
• 🔬 Der Elektron und das Proton im Wasserstoffatom interagieren durch die Coulomb-Wechselwirkung, die das Elektron in Richtung des Protons zieht.
• 🚀 Die Quantenmechanik verhindert, dass das Elektron in das Proton zusammenfällt, da die Unsicherheit der Position und der Impuls die Kollapsenergie verhindert.
• 🤔 Die Elektronenbahnen im Wasserstoffatom können durch Absorption von Photonen in höhere Energiezustände übergehen, was jedoch metastabil ist und schnell zurück in den Grundzustand übergeht.
• 🔄 Die Quantenmechanik erfordert, dass man die genaue Position eines Teilchens nicht bestimmen kann, ohne seine Impulsinformation zu verlieren.
• 🌐 Die sogenannte 'Quantenverschmierung' ist ein fundamentales Merkmal der Quantenmechanik, das die Lokalisierung eines Teilchens verhindert.
• 📏 Die Masse des Protons ist 1836-mal größer als die des Elektrons, was bedeutet, dass das Proton weniger 'verschmiert' ist und als fixer Punkt im Koordinatensystem betrachtet werden kann.
• 📚 Die Wellenfunktion (PSI) ist eine komplexe Funktion von Raum und Zeit und bildet die Grundlage für die Beschreibung des Zustands eines Quantensystems.
• 🧩 Die Hamilton-Operatoren in der Quantenmechanik beschreiben die Energie des Systems und ermöglichen die Lösung des Schrödinger-Gleichungen für Energieeigenzustände.
• 🔗 Die Lösung des Schrödinger-Gleichungs für das Wasserstoffatom liefert die Energieeigenwerte und die zugehörigen Wellenfunktionen, die die möglichen Zustandsbeschreibungen des Elektrons darstellen.

Q & A

Was ist der Grund, warum der Elektron nicht in den Proton fällt?

Der Grund, warum das Elektron nicht in den Proton fällt, liegt in der Quantenmechanik. Die Unsicherheitsrelation zwischen Ort und Impuls verhindert, dass das Elektron exakt lokalisiert werden kann, was eine Kollapse des Elektrons verhindert.

Was ist die Bedeutung von 'Quantenfuzziness' im Kontext des Wasserstoffatoms?

Quantenfuzziness bezieht sich auf die Unsicherheit der Position eines Quantenteilchens, wie das Elektron im Wasserstoffatom. Diese Unsicherheit verhindert, dass das Elektron in den Proton kollabiert.

Wie wird die Energie des Elektrons im Wasserstoffatom quantisiert?

Die Energie des Elektrons im Wasserstoffatom wird durch die Lösung des zeitunabhängigen Schrödinger-Gleichungen quantisiert, was zu diskreten Energieniveaus führt, die als Energieeigenwerte bezeichnet werden.

Was ist die Rolle des Planckschen Wirkungsquantums (h-bar) in der Quantenmechanik?

Das Plancksche Wirkungsquantum (h-bar) ist eine fundamentale Konstante in der Quantenmechanik, die die Beziehung zwischen Energie und Frequenz sowie zwischen Impuls und Ort definiert.

Wie wird die Bewegung des Elektrons im Wasserstoffatom mathematisch beschrieben?

Die Bewegung des Elektrons im Wasserstoffatom wird durch die WellenfunktionPsi beschrieben, die eine komplexe Funktion von Ort und Zeit ist.

Was ist der Hamilton-Operator und welche Rolle spielt er in der Schrödinger-Gleichung?

Der Hamilton-Operator ist eine Differentialoperator, die die Energie eines Systems in Abhängigkeit von Ort und Impuls darstellt. Er ist zentral in der Schrödinger-Gleichung, die die zeitabhängige Entwicklung der Wellenfunktion beschreibt.

Wie wird die kinetische Energie im Rahmen der Quantenmechanik dargestellt?

In der Quantenmechanik wird die kinetische Energie durch den Betrag des Impulsoperators, das heißt durch die quadrierte Gradientenableitung der Wellenfunktion, dargestellt.

Was ist die Bedeutung des magnetischen Quantenzahlen im Zusammenhang mit dem Wasserstoffatom?

Die magnetische Quantenzahl ist ein Index, der die Richtung des Orbitals im Raum beschreibt. Sie ist ein wichtiger Parameter in der Lösung des Schrödinger-Gleichungs für das Wasserstoffatom.

Wie wird die Potentialenergie des Elektrons im Wasserstoffatom berechnet?

Die Potentialenergie des Elektrons im Wasserstoffatom wird durch die klassische Coulomb-Potentialformel berechnet, die die elektrostatische Wechselwirkung zwischen dem Elektron und dem Proton beschreibt.

Was sind 'Energieeigenzustände' und wie sind sie mit der Lösung des Schrödinger-Gleichungs verbunden?

Energieeigenzustände sind spezielle Lösungen der Wellenfunktion, die sich nicht im Raum, sondern nur in der komplexen Ebene verändern. Sie sind mit diskreten Energieniveaus verbunden, die durch die Lösung des zeitunabhängigen Schrödinger-Gleichungs gefunden werden.

Wie wird die sphärische Symmetrie des Wasserstoffatoms in der Lösung des Schrödinger-Gleichungs ausgenutzt?

Die sphärische Symmetrie des Wasserstoffatoms wird durch die Verwendung von sphärischen Koordinaten (R, Theta, Phi) ausgenutzt, was die Lösung des Schrödinger-Gleichungs erleichtern kann.

Outlines

00:00

🌌 Grundzustand des Wasserstoffatoms

In diesem Absatz wird das Wasserstoffatom im Grundzustand beschrieben, bestehend aus einem Protonen und einem Elektron. Der Elektron ist so nah am Protonen wie möglich, bis die Quantenunsicherheit einsetzt. Es wird eine Photon verschossen, um die Reaktion zu beobachten. Das Elektron wird in einen höheren Energiezustand versetzt, der jedoch metastabil ist und bald zurück in den Grundzustand übergeht, wobei das ursprüngliche Photon zurückgeworfen wird. Dies zeigt die Erhaltung der Energie. Es wird auch ein weiteres Photon verschossen, das einen Zustand mit Rotationsmoment aufweist. Der Absatz endet mit einer Meditation über die Situation, die die Verbindung zwischen der Welt der Experimente und der seltsamen Welt der Quantenmechanik hervorhebt.

05:01

🔬 Quantenmechanik und das Wasserstoffatom

Dieser Absatz behandelt die Frage, warum das Elektron nicht in das Protonen fällt, wenn man beide als Punktteilchen modelliert und Maxwells Gleichungen anwendet. Es wird erklärt, dass die Quantenmechanik verhindert, dass das Elektron in das Protonen fällt, da die Unsicherheit der Position und der Impuls des Elektrons die Kollapseinheit verhindert. Es wird auch die Bedeutung der Masse des Protons im Vergleich zum Elektron diskutiert, die die Quantenunsicherheit reduziert. Der Absatz endet mit einer Erklärung der Koordinatensysteme, die für das Wasserstoffatom verwendet werden, insbesondere die sphärischen Koordinaten R, Theta und Phi.

10:02

📚 Wellenfunktion und Wahrscheinlichkeitsdichte

In diesem Absatz werden die grundlegenden Konzepte der Quantenmechanik wie die Wellenfunktion PSI, die Wahrscheinlichkeitsdichte und die reduzierte Planck-Konstante h-bar vorgestellt. Die Wellenfunktion ist eine komplexe Funktion von Raum und Zeit, die die Wahrscheinlichkeitsdichte beschreibt, indem sie die Amplitude des Wellenfunktions quadriert. Die reduzierte Planck-Konstante ist ein fundamentales Element in der Quantenmechanik, das die Beziehung zwischen Energie und Frequenz, Impuls und Raum definiert. Der Absatz führt auch die Schrödinger-Gleichung ein, die die Beziehung zwischen dem Hamilton-Operator und der Wellenfunktion beschreibt und die Grundlage für die Berechnung von Energiezuständen bildet.

15:04

🔗 Energieeigenzustände und Hamilton-Operator

Dieser Absatz erklärt die Energieeigenzustände und wie sie mit dem Hamilton-Operator in Verbindung stehen. Es wird gezeigt, dass die Anwendung des Energieoperators auf einen Energieeigenzustand die Wellenfunktion nur überall im Raum skaliert. Dies wird mit dem klassischen Eigenvektor-Eigenwert-Problem aus der linearen Algebra verglichen. Der Absatz führt dann den Hamilton-Operator für das Wasserstoffatom ein, der die kinetischen und potentialen Energien des Elektrons umfasst. Die kinetischen Energien werden durch den Impulsoperator dargestellt, der aus der klassischen Physik bekannt ist, und die potentielle Energie wird durch die Coulomb-Potentiale des Elektrons und Protons beschrieben.

🧲 Hamilton-Operator und Laplace-Operator

In diesem Absatz wird der Hamilton-Operator für das Wasserstoffatom vollständig formuliert, der sowohl die kinetischen als auch die potentiellen Energien des Elektrons berücksichtigt. Es wird erklärt, dass die Lösung der zeitunabhängigen Schrödinger-Gleichung die Bestimmung der Energieeigenzustände und der entsprechenden Wellenfunktionen erfordert. Der Laplace-Operator wird in sphärischen Koordinaten geschrieben und in die Gleichung eingesetzt, um eine dreidimensionale partielle Differentialgleichung für die Wellenfunktion PSI zu erhalten. Dies ist ein komplexer, aber fundamentaler Schritt in der Lösung der Schrödinger-Gleichung für das Wasserstoffatom.

Keywords

💡Hydrogenatom

Ein Hydrogenatom besteht aus einem Protonen und einem Elektron. Im Grundzustand ist das Elektron so nah wie möglich am Protonen, bis die Quantenmechanik 'Quantenunschärfe' eintritt. Im Video wird das Verhalten des Elektrons im Hydrogenatom im Grundzustand und nach dem Absorption von Photonen diskutiert.

💡Quantenmechanik

Quantenmechanik ist eine Theorie in der Physik, die das Verhalten von Teilchen auf sehr kleinen Skalen beschreibt. Im Video wird gezeigt, wie die Quantenmechanik die Stabilität des Elektrons im Hydrogenatom erklärt, da es nicht in den Protonen fallen kann, sondern durch die Unschärfe der Wellenfunktion 'gefangen' bleibt.

💡Wellenfunktion

Die Wellenfunktion ist eine komplexe Funktion, die die Quantenzustände eines Teilchens beschreibt. Im Video wird die Wellenfunktion verwendet, um die Position des Elektrons im Hydrogenatom zu beschreiben und wie sie durch die Absorption von Photonen verändert wird.

💡Schroedingersche Gleichung

Die Schroedingersche Gleichung ist eine zentrale Gleichung in der Quantenmechanik, die die zeitabhängige Entwicklung der Wellenfunktion eines Teilchens beschreibt. Im Video wird die Gleichung verwendet, um die Energiezustände des Elektrons im Hydrogenatom zu bestimmen.

💡Energieeigenzustände

Energieeigenzustände sind Zustände, in denen ein Teilchen eine bestimmte, diskrete Energie hat. Im Video werden diese Zustände verwendet, um die verschiedenen Energieniveaus des Elektrons im Hydrogenatom zu beschreiben, die durch das Absorbieren von Photonen erreicht werden können.

💡Quantenunschärfe

Die Quantenunschärfe ist ein Grundprinzip der Quantenmechanik, das besagt, dass die Position und der Impuls eines Teilchens nicht gleichzeitig genau bestimmt werden können. Im Video wird dies verwendet, um zu erklären, warum das Elektron nicht in den Protonen fallen kann.

💡Hamilton-Operator

Der Hamilton-Operator ist in der Quantenmechanik die Energie des Systems, geschrieben in Form von Position und Impuls. Im Video wird er verwendet, um die Schroedingersche Gleichung für das Hydrogenatom zu konstruieren und die Energiezustände zu berechnen.

💡Laplacian

Das Laplacian ist eine Differentialoperator, der die Konvexiät einer Funktion in drei Dimensionen misst. Im Video wird es verwendet, um den kinetischen Energie-Operator in der Schroedingerschen Gleichung zu definieren, was wichtig ist, um die Bewegung des Elektrons im Hydrogenatom zu beschreiben.

💡Reduzierte Planck-Konstante

Die reduzierte Planck-Konstante, h-bar, ist ein fundamentales Maß in der Quantenmechanik, das die Beziehung zwischen Energie und Frequenz sowie zwischen Impuls und Raum definiert. Im Video wird h-bar verwendet, um die Gleichungen in der Schroedingerschen Gleichung zu skalieren.

💡Sphärische Koordinaten

Sphärische Koordinaten sind ein Koordinatensystem, das für Probleme mit kugelsymmetrischer Geometrie verwendet wird. Im Video werden sphärische Koordinaten verwendet, um die Laplacian und die Wellenfunktion des Elektrons im Hydrogenatom zu beschreiben.

Highlights

Quantum fuzziness prevents the electron from collapsing into the proton in a hydrogen atom.

Electron transitions to higher energy states upon photon absorption, exemplified by the 2p0 state.

Metastable states, like the 2p0, are temporary and the electron returns to the ground state, releasing a photon.

The electron's angular momentum is demonstrated in the 2p1 state, showcasing quantum mechanics' intricacies.

The paradox of the electron not falling into the proton is explained by quantum mechanics' uncertainty principle.

Localization of a quantum particle increases its momentum, preventing it from collapsing due to energy requirements.

The reduced mass of the electron is used in the Hamiltonian for the hydrogen atom, accounting for the proton's finite mass.

Spherical coordinates are chosen for the hydrogen atom due to its spherical symmetry.

The wave function, probability density, and reduced Planck's constant are fundamental to quantum mechanics.

The Schrödinger equation is the key to understanding the relationship between energy, momentum, space, and time in quantum systems.

Energy eigenstates, or stationary states, are solutions to the time-independent Schrödinger equation.

The Hamiltonian operator for the hydrogen atom combines kinetic and potential energy terms.

The quantum mechanical momentum operator is derived from the classical momentum, using the reduced Planck's constant.

The potential energy operator in the Hamiltonian is based on the Coulomb potential between the electron and proton.

The time-independent Schrödinger equation is solved for energy eigenstates to find the allowable energy levels of the hydrogen atom.

The Laplacian in spherical coordinates is used to transform the Schrödinger equation into a solvable form.

The solution to the Schrödinger equation for the hydrogen atom reveals the quantum mechanical behavior of electrons.

Transcripts

foreign

look at this beautiful hydrogen atom in

the ground state there's one proton one

electron and the electron is as close as

it can be to the proton until Quantum

fuzziness kicks in and the electrons

kind of in this wave function of

positions and you don't know exactly

where it is but it's something like this

let's shoot a photon at this and see

what happens

look at this it's a two zero zero State

very nice so now the electron is a bit

farther out a bit less bound to the

proton in a higher energy State relative

to the ground state but be careful this

is only metastable it's going to pop

back down soon so any minute now it's

going to pop into the ground state

oh there it goes and look we got our

Photon back did you see that flash of

light

conservation of energy very nice let's

put another Photon into it and see what

happens

hey that's a two one zero State nice you

know that one has some angular momentum

oh there it goes

let's take a moment to meditate on this

situation

[Music]

foreign

[Music]

[Music]

we'll Begin by examining our atom in its

most relaxed form this dazzling little

pattern is one of Nature's most abundant

most ancient motifs

but there's a deep mystery here why is

it that the electron doesn't just fall

into the proton

if you model the electron and the proton

as Point particles and apply Maxwell's

equations you'll find that the electron

will radiate out its energy and will end

up falling into the proton in just a few

nanoseconds

but there's hydrogen out in space that's

like billions of years old so clearly

our math is a little bit off because

hydrogen actually doesn't Decay

instantly so what is it that stops the

collapse

have you ever tried to catch a Quantum

particle

imagine you have one and you've caught

it you're pinching it between your

finger and your thumb and you squeeze it

really tight so you know just exactly

where it is you know it's positioned

with perfect precision

oh well by quantum mechanics now you no

longer know its momentum and so it

escapes

in quantum mechanics you actually can't

perfectly localize a single particle

you can try but it takes a lot of energy

and the tighter you squeeze it the more

you localize it the more energy it takes

if you think about it a proton is

pulling in the electron the electrons

this Quantum particle it wants to

collapse all the way but eventually

there's a point where the quantum

fuzziness makes it so that the

uncertainty and momentum keeps the thing

from falling all the way in

and so you see hydrogen is not just an

atom it's also this portal between the

world of experiment and the very strange

and unusual world of quantum mechanics

that bubbles up into our world wait hold

up so the electron is a Quantum particle

and it's all fuzzy but the proton is

just this point-like thing how does that

make sense

well it's because the proton is about

1836 times as massive as the electron so

just to put this into perspective the

difference in Mass between an electron

and a proton is the difference between

an elephant

and 1836 elephants so the proton is very

very massive because it's so much more

massive it's less fuzzy it is still

fuzzy if you look very closely at it

it's fuzzy but it's much less fuzzy

because there's this inverse

relationship between distance and mass

when it comes to quantum mechanics

because the proton is so much more

massive than the electron we can do all

of our analysis by assuming that the

proton will be at the center of our

coordinate system and that it doesn't

move it just stays put and the electron

does whatever quantum mechanical cloudy

wavy stuff it does okay all right let's

talk about coordinates normally I like

to use Cartesian coordinates X Y and Z

but because of the nature of this

problem it has a spherical Symmetry and

so spherical coordinates fit like a hand

in a glove to this problem so we're

going to use these the coordinates R

Theta and Phi one thing I have to point

out I got to be careful here so normally

I use Theta as the angle around the

longitude like the azimuthal angle and I

used Phi for the elevation angle but for

whatever reason physicists working on

the hydrogen atom always use the other

way of defining Theta and Phi and so I'm

going to go along with that convention

but just be aware this is a little bit

different than the convention that I

normally use so just to be really clear

Theta is actually going to be our

elevation angle so that's going to be

the angle that starts off at zero on the

North Pole and then goes down to Pi or

180 degrees at the South Pole and then

Phi is going to be our azimuthal angle

so that's the angle that's going to go

around the equator zero at the Prime

Meridian and then you know it goes

around a full 360 or full 2 pi

okay so now that we've defined our

coordinate system let's define some of

the most important things in quantum

mechanics the first thing is the wave

function so the wave function is this

complex valued function that's a

function of both space and time so the

wave function is given the symbol PSI

and PSI depends in this case on R Theta

Phi and time

closely related to the wave function is

the probability density that is the

thing that if you integrate over some

volume you get the probability that the

particle is going to be in that volume

the probability density is just the

amplitude squared of the wave function

when you take the amplitude squared of a

complex number you get a real number so

the probability density is a real valued

function and it's also a function of

space and time although as we'll see

when we solve any dragon States it's

just a function of space all right and

finally the reduced Planck's constant

this number h-bar you see this

everywhere in quantum mechanics it's

absolutely ubiquitous it's a measurable

quantity it has about the value of

1.05457 times 10 to the minus 34 Joule

seconds this is a very mysterious number

it is what it is and no one knows why it

is it just is and so you'll see this in

many of our equations today it defines

the relationship between energy and

frequency and momentum and space and all

kinds of stuff sort of the quantum scale

of angular momentum or action and by the

way I should mention you know why they

call it h bar it's actually plan's

constant H divided by 2 pi why but so

often you divide by 2 pi that people got

tired of writing divided by 2 pi so then

they just put a bar on the H now

everyone knows that means divide by 2 pi

so we want to figure out what is our

electron up to what does it do and in

order to do that we need an equation

that lets us relate things like momentum

and space and time

and so what we're going to do is we're

going to use the Schrodinger equation

shown here the Schrodinger equation is

just the idea that if the hamiltonian

operator acts on a wave function that's

the same thing as the energy operator

acting on a wave function now there's a

lot of confusion when people first see

hamiltonian operator they're not sure

what that is because it's just a thing

named after some guy so who knows what

it is well what it is is the energy

written in terms of position and

momentum and we'll see in a moment

exactly how to construct the hamiltonian

for the hydrogen atom

the energy operator is in quantum

mechanics it's defined as I H Bar times

partial PSI partial T so in other words

you take the partial derivative of the

wave function in time then you rotate it

90 degrees in the complex plane by

multiplying by I and then you multiply

it by that Quantum scale parameter h-bar

now if you look at this you might be

wondering why is this the energy

operator where does this come from and

the answer is today we're just going to

take this as one of our principles as

one of our assumptions that we're going

to use to build up this theory if you're

interested more in the nature of the

energy operator I'd recommend the book

quantum mechanics and path integrals by

Feynman and hibs this book constructs

quantum mechanics from a pretty

intuitive starting point well relatively

for quantum mechanics and and then they

show that you can basically derive all

of this Schrodinger wave equations from

path integrals now the problem with path

integrals is they're impossible to work

with but they're very nice to imagine so

if you want to learn more about why the

energy operator is what it is check out

that book but today we're just going to

take the energy operator for granted and

we're going to continue forward now when

we solve the Schrodinger equation we're

not just interested in every possible

wave function as a function of space and

function of time we're actually

particularly interested in these things

called Energy eigenstates they're also

known as stationary States I like to

think of them as resonant mode although

that's maybe kind of an analogy but I

think it's a good one

so an energy eigenstate is a wave

function that doesn't move except it

just rotates in the complex plane

so in other words you can break it up so

the wave function is a function of space

and time

can be thought of as the wave function

as a function of space

times this time parameter which just

swings around in the complex plane and

the frequency of how much it swings

around has to do with the energy of the

wave function

so when we solve for the time

independent Schrodinger equation what

that means is we want to figure out what

are all the patterns what are all the

different wave functions as a function

of space and then what are the

corresponding energy levels by the way

the energy levels are also called Energy

eigenvalues

the deal with all this eigen stuff

anyway well if you've studied linear

algebra then you'll be familiar with

eigenvector and eigenvalue problems

normally you'll have some kind of linear

transformation and then there are

specific vectors that are just uniformly

scaled by that transformation and the

amount to which they're scaled is the

eigenvalue and the vectors themselves

are eigenvectors and eigen I think it

comes from some German word meaning own

or self or like related to the thing

it's confusing terminology admittedly

but let's just apply the energy operator

to an energy eigenstate and see how we

can draw that parallel between

eigenvector eigenvalue problems and this

whole thing about eigenstates if we

apply our energy operator to psi so we

do I H bar partial PSI partial T and we

substitute in our wave function which is

our energy eigenstate where we have a

spatial part and a time part and then

we'd work out the derivatives what we

find is that the energy operator

basically amounts to just scaling the

wave function by a constant everywhere

in space

and so you'll notice that this seemingly

simple looking equation E hat PSI equals

e PSI it's actually pretty profound and

this should look a lot like your classic

you know Matrix times eigenvector equals

eigenvalue times eigenvector equation

from linear algebra by the way that's

not a coincidence if you've studied

structural engineering and you've

calculated resonant modes and

frequencies you'll see there's really a

one-to-one parallel between that

situation and what's going on here today

okay so just to recap the time dependent

Schrodinger equation that is the general

the real for real Schrodinger equation

is the equation that the hamiltonian

operator acting on a wave function is

the same as the energy operator acting

on a wave function and that lets us

relate momentum and space and time and

we can derive the governing equations of

our wave function

if we restrict our attention to solving

for these energy eigenstates which you

can imagine is resonant modes or the

ways in which the equation rings then we

end up with the time independent

Schrodinger equation in which the energy

operator is replaced by a constant that

constant of course depends on the

particular energy eigenstate we're

looking at some of them will have higher

energies some of them will have lower

energies but in any case we can regard

that energy level as an eigenvalue of

the hamiltonian operator acting on our

wave function

let's construct the hamiltonian for the

hydrogen atom to do that we need to add

the electrons kinetic and potential

energy

first let's start with the kinetic

energy

from classical non-relativistic physics

we know that the kinetic energy T is

equal to one-half MV squared where m is

the mass of the particle and V is the

velocity we also know from classical

physics that momentum p is mass times

velocity

therefore if you just rearrange those

equations you can prove to yourself that

the kinetic energy is the momentum

squared divided by twice the mass

and in quantum mechanics we're going to

use that exact same idea except we're

going to make the momentum a Quantum

thing how do we do that well we use the

quantum mechanical momentum operator

so the momentum operator P hat acting on

the wave function PSI is negative i h

Bar times the gradient of PSI

now if we use our formula from classical

physics that the kinetic energy is

momentum squared divided by twice the

mass then we can derive the quantum

mechanical kinetic energy Operator by

applying the momentum operator twice and

dividing by twice the mass

when we do that we find that the kinetic

energy operator t-hat applied to a wave

function PSI gives you negative H bar

squared over 2m times the laplacian of

PSI

and what that means intuitively is that

if you're going to take the kinetic

energy of a wave function you look at

its laplacian the laplacian is basically

the concavity in three dimensions it's

like a second derivative but adding up

along all the different second

derivative and X Plus second derivative

and Y plus second derivative and z and

then you multiply that concavity by H

bar squared over 2m and then you take

the minus sign of that

so you know earlier we were talking

about how it takes energy to localize a

particle the more you squeeze it the

more it sort of pushes back well we can

mathematically encode that in this

equation with the kinetic energy

operator right because you think about

it the more you pinch a particle the

more you're increasing its laplacian you

know the laplacian in a way is sort of

the extent to which the wave function is

pinched right it's the Divergence of the

gradient so the more you pinch it the

more this t-hat term increases

now if we look at the potential energy

from the electron and proton coulomb

potential so in other words just the

regular old static electricity Coulomb's

law we can see that the potential energy

operator V acting on our wave function

PSI is just the Classic minus Elementary

charge squared over 4 pi times the

permittivity of free space times the

radial coordinates all acting on our

wave function PSI

and so what that means is that there's

going to be a potential energy term in

our equation that drops off as 1 over r

but notice there's a minus sign on this

potential energy and so actually a

bigger magnitude means it's more

negative

negative energy in this context just

means that it's less than zero so if the

electron and proton are infinitely far

away let's call that zero then the

coulomb potential is negative because it

represents a kind of energy dead you'd

have to put energy into the hydrogen

atom in order to get the electron out

and so actually this one over R scaling

of the electrostatic potential is going

to tend to pull the electron in to the

proton

and so when we add the kinetic and

potential energy terms together in our

hamiltonian what we're describing when

we do that is that balance of energies

we were talking about earlier between

the electron getting pulled into the

proton but also that quantum mechanical

fuzziness that kinetic energy keeping

the electron from falling all the way in

and so we can finally write our

hamiltonian operator H hat acting on PSI

as negative H bar squared divided by

twice the mass times the laplacian of

our wave function minus the fundamental

charge squared divided by 4 Pi

permittivity of free space r times PSI

okay maybe it looks like a lot if this

is the first time you've seen it but all

that is to say the energy of the

electron has a kinetic term and it has

an electrostatic potential term

now something I should mention here is

that we want to actually use something

called the reduced mass of the electron

so this is basically the same thing as

the electron Mass it's like a little

tiny bit less like a part in a thousand

less kind of thing and what that does is

it lets us account a little bit for the

fact that the proton actually has finite

Mass it's not infinitely massive this

idea comes from orbital mechanics I

believe is where this first comes from

but for our purposes today basically the

main advantage is it lets us replace the

letter M with the letter mu because

we're going to need M later on when we

get to the magnetic quantum number

all right well now that we have our

hamiltonian we can plug it into the time

independent Schrodinger equation that is

H hat PSI equals e PSI where e is the

energy eigenvalue and PSI is an energy

eigenstate

let's massage this equation a little bit

we'll move the E side term on over to

the left side of the equation we'll

cancel out some minus signs and we get

this pretty looking equation that the

laplacian of PSI plus 2 mu over H bar

squared times e squared over 4 Pi

Epsilon not R plus e times PSI equals

zero

so up until this point we've used

physics and this idea of energy

operators and Schrodinger equation and

hamiltonian so we've compiled this

equation but now solving this equation

is an exercise in math

because we can just look at it as a

three-dimensional partial differential

equation and ask what are the functions

PSI that satisfy this equation

So to that end the first thing that we

should do is write out the laplacian in

terms of partial derivatives of PSI with

respect to R Theta and Phi

now here's the thing uh so earlier I

mentioned that using spherical

coordinates was going to help us out

because of the spherical nature of the

problem and that is true spherical

coordinates are very nice trust me we do

want to use them but there's one way in

which they're not so nice and that is

when you write out the laplacian it's

quite an expression anyway I'm not going

to go into the whole derivation of this

now but if you just look up laplacian

written in spherical coordinates you'll

see this expression it's a bit

complicated but it is what it is you

know no matter how fun it is it is what

it is okay now all we have to do is take

our expression for the laplacian and put

it into that equation and what we end up

with is a three-dimensional partial

differential equation for PSI as a

function of the variables R Theta and

Phi

wow look at this thing oh what a mess

but there it is this is a beautiful

equation in a way

so let's solve it let's solve it for PSI

how hard can it be