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

Summary

视频脚本通过氢原子模型深入探讨了量子力学的基本原理。首先，介绍了氢原子的基态，其中电子围绕质子运动，但由于量子不确定性，电子的位置是概率性的。接着，通过向氢原子发射光子，展示了电子跃迁到较高能级的过程，并最终返回基态，释放光子，体现了能量守恒。视频还讨论了为何电子不会落入质子，引入了量子力学中的不确定性原理。此外，通过比较电子和质子的质量差异，解释了为何可以假设质子固定不动。最后，使用球坐标系和薛定谔方程，建立了氢原子的哈密顿算符，并推导了时间独立的薛定谔方程，为解决电子的行为提供了数学基础。整个脚本是对量子力学核心概念的精彩阐释，旨在激发观众对这一神秘领域的兴趣。

Takeaways

• 🚀 氢原子的基态包含一个质子和一个电子，电子尽可能地靠近质子，直到量子不确定性起作用，电子的位置变得模糊。
• 🔬 当向氢原子发射光子时，电子会被激发到一个更高的能级，但这个状态是亚稳态，电子很快会回落到基态并释放光子，这是能量守恒的体现。
• 🌌 电子作为量子粒子，其位置和动量不能同时被精确知晓，这种量子不确定性阻止了电子落入质子中，从而解释了氢原子为何不会衰变。
• 📍 在量子力学中，我们通常使用球坐标系（R, Θ, Φ）来描述具有球对称性的问题，如氢原子。
• 📈 波函数（Ψ）是一个复数函数，它依赖于空间和时间，概率密度是波函数的模长平方，用于描述粒子出现在某个体积内的概率。
• 🎓 约化普朗克常数（ℏ）在量子力学中非常重要，它关联了能量、频率、动量和空间等量子尺度的物理量。
• 🧲 薛定谔方程是量子力学中的基本方程，它关联了哈密顿算符（能量算符）和波函数，用于描述量子系统的时间演化。
• ⚙️ 哈密顿算符结合了电子的动能和与质子的电势能，动能算符与波函数的二阶导数（拉普拉斯算子）有关，而电势能算符与1/R有关。
• 🔧 通过求解时间独立的薛定谔方程，我们可以得到氢原子的能量本征态（或称定态），这些状态描述了电子在不同能级上的概率分布。
• 🧮 球坐标系下的拉普拉斯算子形式较为复杂，但它是求解氢原子波函数的关键数学工具。
• 🔬 电子和质子之间的相互作用通过库仑定律描述，电子被质子吸引，但由于量子力学的不确定性原理，电子不会完全落入质子中。
• 📚 解决氢原子的薛定谔方程是一个数学问题，需要找到满足特定偏微分方程的函数，这些函数将给出电子在不同能级上的概率分布。

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Outlines

00:00

🌌 氢原子的量子行为

本段介绍了氢原子的基本结构，包括质子、电子以及电子在量子力学中的波动性。通过发射光子观察电子从基态到激发态的跃迁，并讨论了量子力学中的不确定性原理，即无法同时准确知道粒子的位置和动量。此外，还探讨了为何电子不会落入质子，以及量子力学与经典力学在描述微观世界时的差异。

05:01

📐 量子力学的数学基础

这一段深入讨论了量子力学中的几个关键概念，包括波函数、概率密度、约化普朗克常数以及薛定谔方程。解释了波函数是如何描述粒子在空间和时间中的状态，以及如何通过波函数的幅度平方得到粒子出现在某一体积内的概率。还介绍了哈密顿算符和能量算符，以及它们在量子力学中的重要性。

10:02

🧲 氢原子的哈密顿量构建

本段内容聚焦于构建氢原子的哈密顿量，这是量子力学中描述系统总能量的算符。首先介绍了动能的量子力学表达式，然后讨论了电子和质子之间的库仑势能。通过将动能项和势能项结合起来，得到了氢原子的哈密顿量，这个量描述了电子与质子之间的能量平衡。

15:04

🔍 求解薛定谔方程

最后一段讲述了如何将哈密顿量代入时间独立的薛定谔方程中，并将其转换为一个三维偏微分方程。这个方程用于寻找满足特定条件的波函数，即能量本征态。通过将拉普拉斯算子在球坐标系中表达，并将其代入方程，得到了描述氢原子中电子行为的复杂数学模型。

Keywords

💡氢原子

氢原子是最简单的原子，由一个质子和一个电子组成。在视频中，氢原子被用来展示量子力学的基本原理，如电子不能落入质子中，而是存在于一定的能量状态中。这与经典物理学的预测不符，展示了量子力学与经典物理学的区别。

💡量子力学

量子力学是描述微观粒子行为的物理理论。视频中通过氢原子模型解释了量子力学中的一些基本概念，如波函数、量子不确定性和能量本征态。量子力学揭示了原子和亚原子粒子的奇异行为，与我们日常经验中的物理现象截然不同。

💡波函数

波函数（Ψ）是量子力学中的一个复数函数，它与粒子在空间中的位置和时间有关。视频中提到，波函数的绝对值的平方给出了粒子在某个区域内被发现的概率密度。波函数是理解量子系统状态的基础。

💡能量本征态

能量本征态是量子力学中描述系统可能状态的解，它们对应于特定的能量值。视频中通过氢原子的模型展示了能量本征态的概念，说明了电子在不同能量状态下的行为，以及如何通过吸收和释放光子在不同的能量状态之间跃迁。

💡薛定谔方程

薛定谔方程是量子力学中的基本方程，用于描述量子系统的时间演化。视频中提到了时间依赖和时间独立的薛定谔方程，这些方程帮助我们理解量子系统的行为，尤其是如何通过哈密顿算符来描述系统的能量。

💡哈密顿算符

哈密顿算符是量子力学中用于描述系统能量的算符。它由系统的动能和势能组成。在视频中，哈密顿算符被用来构建氢原子的量子模型，并求解电子的波函数。

💡动量算符

动量算符是量子力学中用来表示粒子动量的算符。它与粒子的波函数相联系，并在薛定谔方程中起到关键作用。视频中通过动量算符解释了量子粒子的动量和位置如何通过不确定性原理相互关联。

💡不确定性原理

不确定性原理是量子力学的一个基本原理，表明不能同时精确知道粒子的位置和动量。视频中通过尝试“捕捉”量子粒子的比喻来解释这一原理，说明了量子系统的固有模糊性。

💡球坐标

球坐标是描述三维空间中点位置的坐标系统，由半径、极角和方位角组成。视频中提到，由于氢原子具有球对称性，使用球坐标可以更方便地解决薛定谔方程。

💡角动量

角动量是描述粒子旋转运动的物理量。在量子力学中，角动量是量子化的，意味着它只能取特定的值。视频中提到了角动量在氢原子的量子态中的作用，以及如何通过不同的角度（如θ和φ）来描述粒子的轨道。

💡普朗克常数

普朗克常数是量子力学中的一个基本常数，它关联了粒子的能量和其频率。视频中提到了约化普朗克常数（ħ），它是普朗克常数除以2π，用于描述量子系统中的能量和动量关系。

Highlights

氢原子的电子在基态时，与质子的距离由量子不确定性决定，电子存在于一种波函数中。

通过向氢原子发射光子，可以观察到电子跃迁到高能级状态，但这种状态是亚稳态，电子会很快跃迁回基态。

氢原子展示了量子力学中电子与质子之间的相互作用，电子不会落入质子中，因为量子不确定性和动量的作用。

量子力学中，无法精确定位一个粒子的位置和动量，这种不确定性随着粒子的局域化程度增加而增加。

质子的质量远大于电子，因此在分析中可以假设质子固定不动，而电子进行量子力学的运动。

使用球坐标系统（R, Θ, Φ）来描述氢原子问题，因为其球对称性。

波函数Ψ是一个复数函数，依赖于空间和时间，是量子力学中描述粒子状态的基本工具。

概率密度是波函数的幅度平方，用于描述粒子出现在某个体积内的概率。

约化普朗克常数ħ是量子力学中的关键常数，关联能量、频率、动量和空间。

薛定谔方程是量子力学中描述粒子状态随时间演化的基本方程。

能量本征态（或称定态）是波函数在空间中不随时间变化，仅在复平面中旋转的状态。

哈密顿算符是能量的量子力学表示，由位置和动量的表达式构成。

氢原子的哈密顿算符由电子的动能项和电子-质子间的库仑势能项组成。

使用约化质量代替电子质量可以更准确地描述氢原子中的电子运动。

通过时间独立的薛定谔方程，可以得到描述电子在不同能量本征值下的空间波函数。

球坐标下的拉普拉斯算符表达式复杂，但在解决氢原子问题时非常有用。

将拉普拉斯算符代入薛定谔方程，得到一个三维偏微分方程，用于求解波函数Ψ。

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