The Hydrogen Atom, Part 1 of 3: Intro to Quantum Physics
Summary
TLDRThis script delves into the quantum mechanics of a hydrogen atom, illustrating its ground state with a proton and electron. It explains the electron's wave function and its transition to higher energy states upon photon absorption, which is temporary due to a return to the ground state, releasing the photon. The script explores the mystery of why the electron doesn't collapse into the proton, introducing the concept of quantum uncertainty and the role of the reduced Planck constant. It also discusses the use of spherical coordinates and the Schrödinger equation to describe the electron's behavior, leading to an exploration of energy eigenstates and the construction of the Hamiltonian for the hydrogen atom.
Takeaways
- 🚀 The script discusses the behavior of a hydrogen atom, focusing on the electron's wave function and its relationship with the proton.
- 🔬 When a photon is shot at the hydrogen atom, the electron moves to a higher energy state, but this is only metastable and it quickly returns to the ground state, releasing the photon.
- 🌌 The script highlights the mystery of why an electron doesn't fall into the proton, contrary to what classical physics would predict.
- 📚 It explains the Heisenberg Uncertainty Principle, which states that the more precisely the position of a quantum particle is known, the less precisely its momentum is known, and vice versa.
- 🧲 The script uses the analogy of trying to 'squeeze' a quantum particle to illustrate the principle of quantum uncertainty and the energy required to localize a particle.
- ⚛ The difference in mass between an electron and a proton is emphasized, with the proton being significantly more massive and thus less 'fuzzy' in its quantum state.
- 📏 Spherical coordinates (R, Theta, Phi) are chosen for the problem due to the spherical symmetry of the hydrogen atom, with a note on the unconventional use of these variables in physics.
- 🌐 The wave function (Ψ) is described as a complex-valued function of space and time, central to quantum mechanics and related to the probability density of finding a particle.
- 🔑 The reduced Planck's constant (h-bar) is introduced as a fundamental constant in quantum mechanics, relating energy, frequency, and momentum.
- 💡 The Schrodinger equation is presented as the key equation for determining the behavior of quantum systems, relating the Hamiltonian operator to the energy operator.
- 🔍 The construction of the Hamiltonian for the hydrogen atom is detailed, combining the kinetic and potential energies to describe the balance of forces acting on the electron.
Q & A
What is the ground state of a hydrogen atom?
-The ground state of a hydrogen atom is the lowest energy state where the electron is as close as quantum mechanics allows to the proton without collapsing into it.
What happens when a photon is absorbed by an electron in the ground state of a hydrogen atom?
-When a photon is absorbed, the electron transitions to a higher energy state, such as the 2p0 state, which is less bound to the proton and farther out. This state is metastable and the electron will eventually return to the ground state, releasing the photon.
Why doesn't the electron in a hydrogen atom fall into the proton?
-The electron doesn't fall into the proton due to the Heisenberg uncertainty principle, which states that you cannot simultaneously know the exact position and momentum of a particle. The 'quantum fuzziness' prevents the electron from collapsing into the proton.
What is the significance of the reduced Planck's constant (h-bar) in quantum mechanics?
-The reduced Planck's constant (h-bar) is a fundamental constant in quantum mechanics that relates energy to frequency and momentum to position. It is ubiquitous in quantum equations and defines the scale of quantum phenomena.
What is the role of the Schrödinger equation in understanding the behavior of an electron in a hydrogen atom?
-The Schrödinger equation is used to calculate the wave function of the electron, which describes the probability distribution of the electron's position in space and time. It is essential for determining the energy eigenstates of the electron.
What are energy eigenstates and why are they important in quantum mechanics?
-Energy eigenstates are wave functions that represent the stationary states of a quantum system, where the only change over time is a phase rotation. They are important because they correspond to definite energy levels of the system.
How is the kinetic energy operator in quantum mechanics related to the momentum operator?
-The kinetic energy operator in quantum mechanics is derived from the momentum operator by applying it twice and dividing by twice the mass of the particle. It represents the kinetic energy as the negative of the Laplacian of the wave function multiplied by h-bar squared over 2m.
What is the potential energy operator for the electron-proton system in a hydrogen atom?
-The potential energy operator for the electron-proton system is given by the negative of the product of the elementary charge squared, the radial coordinate r, and the wave function, divided by 4 Pi times the permittivity of free space.
What is the Hamiltonian operator and how is it constructed for a hydrogen atom?
-The Hamiltonian operator is the total energy operator in quantum mechanics, which includes both the kinetic and potential energy of a system. For a hydrogen atom, it is constructed by combining the kinetic energy operator and the potential energy operator due to the Coulomb force between the electron and proton.
Why is the reduced mass used in the Hamiltonian for the hydrogen atom instead of the electron mass?
-The reduced mass is used to account for the fact that the proton has finite mass. It allows for a more accurate description of the system and simplifies the equations by avoiding the need to consider the motion of the proton.
What is the significance of the spherical symmetry in the problem of the hydrogen atom and how does it affect the choice of coordinates?
-The spherical symmetry of the hydrogen atom means that the potential energy is the same in all directions. This makes spherical coordinates, which naturally reflect this symmetry, the most suitable choice for solving the Schrödinger equation for the hydrogen atom.
Outlines
🌌 Quantum Mechanics of a Hydrogen Atom
This paragraph introduces the quantum behavior of a hydrogen atom in its ground state, with a focus on the electron's position and its interaction with a proton. It explains the electron's wave function and the uncertainty principle, which prevents the electron from collapsing into the proton. The script also demonstrates what happens when a photon is absorbed by the electron, causing it to move to a higher energy state, and then returns to the ground state, releasing the photon. This process illustrates the conservation of energy and the quantum nature of light and matter.
🔬 The Mystery of Hydrogen's Stability
The script delves into the paradox of why an electron in a hydrogen atom doesn't fall into the proton, despite classical predictions. It highlights the role of quantum mechanics in preventing the collapse and introduces the Heisenberg Uncertainty Principle, which states that the more precisely the position of a particle is measured, the less precisely its momentum can be known, and vice versa. The paragraph also contrasts the quantum 'fuzziness' of the electron with the relative point-like nature of the proton due to its much greater mass.
📏 Spherical Coordinates and Quantum Mechanics
The narrator discusses the choice of spherical coordinates (R, Theta, Phi) for analyzing the hydrogen atom due to its spherical symmetry. It clarifies the convention used in physics for these coordinates, which differs from the usual mathematical convention. The paragraph introduces key quantum mechanical concepts, including the wave function (PSI), probability density, and the reduced Planck's constant (h-bar), which are fundamental to solving the quantum behavior of particles.
🧬 Schrodinger Equation and Energy Eigenstates
This section explains the importance of the Schrodinger equation in quantum mechanics, which connects the wave function's energy to its spatial and temporal properties. The focus is on energy eigenstates, or stationary states, which are solutions to the time-independent Schrodinger equation. These states represent the natural frequencies at which the system oscillates and are likened to resonant modes in engineering. The paragraph also touches on the concept of eigenvalues and eigenvectors in the context of quantum mechanics.
🚀 Constructing the Hamiltonian for the Hydrogen Atom
The script outlines the process of constructing the Hamiltonian operator for the hydrogen atom, which combines the kinetic and potential energy of the electron. It describes how the kinetic energy operator is derived from the quantum mechanical momentum operator and how the potential energy is determined by the Coulomb potential between the electron and proton. The paragraph also introduces the reduced mass concept, which accounts for the proton's finite mass in the calculations.
🔍 Solving the Time-Independent Schrodinger Equation
The final paragraph details the process of solving the time-independent Schrodinger equation for the hydrogen atom. It presents the equation in terms of the laplacian operator and the energy eigenvalue, which simplifies to a three-dimensional partial differential equation. The paragraph emphasizes the mathematical challenge of solving this equation and sets the stage for finding the quantum states of the hydrogen atom.
Mindmap
Keywords
💡Hydrogen Atom
💡Quantum Mechanics
💡Wave Function
💡Quantum Fuzziness
💡Schrodinger Equation
💡Energy Eigenstates
💡Hamiltonian Operator
💡Reduced Planck's Constant
💡Laplacian
💡Spherical Coordinates
💡Coulomb's Law
Highlights
Introduction to the quantum behavior of a hydrogen atom in its ground state, illustrating the electron's position as a wave function.
Demonstration of electron excitation to a higher energy state by shooting a photon at the hydrogen atom.
Explanation of the electron's return to the ground state and the conservation of energy through photon emission.
Discussion on why the electron doesn't fall into the proton, challenging classical physics with quantum mechanics.
Introduction of the Heisenberg Uncertainty Principle and its role in preventing electron-proton collapse.
Illustration of the difference in mass between an electron and a proton, using the analogy of an elephant to 1836 elephants.
Introduction to spherical coordinates as the most suitable system for analyzing the hydrogen atom due to its spherical symmetry.
Clarification of the convention used for Theta and Phi in spherical coordinates, different from the usual geographical convention.
Introduction of the wave function PSI, its significance in quantum mechanics, and its relationship to probability density.
Explanation of the reduced Planck's constant, h-bar, and its fundamental role in quantum mechanics.
Introduction to the Schrödinger equation as the key to understanding the relationship between momentum, space, and time in quantum systems.
Discussion on energy eigenstates and their importance in solving the Schrödinger equation for the hydrogen atom.
Construction of the Hamiltonian for the hydrogen atom, combining kinetic and potential energy terms.
Derivation of the quantum mechanical kinetic energy operator from classical physics concepts.
Introduction of the potential energy operator based on Coulomb's law and its significance in the hydrogen atom model.
Final formulation of the Hamiltonian operator in terms of the wave function PSI, setting the stage for solving the Schrödinger equation.
Transformation of the time-independent Schrödinger equation into a three-dimensional partial differential equation in spherical coordinates.
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
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foreign
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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
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