Complex Numbers in Quantum Mechanics
Summary
TLDRThis video script delves into the intricacies of quantum mechanics, particularly the use of complex numbers to represent wave phenomena. It explains the two-dimensional nature of complex numbers as a generalization of positivity and negativity, illustrating their relevance through the analogy of waves. The script further explores how complex numbers facilitate the understanding of wave interference, Fourier analysis, and quantum systems like the quantum harmonic oscillator. It also touches on the profound implications of complex numbers in quantum electrodynamics, leaving viewers intrigued by the deep connections between mathematics and the physical world.
Takeaways
- 🧠 Quantum mechanics is complex due to the counterintuitive phenomena it describes and the use of complex numbers.
- 🔢 Complex numbers are two-dimensional, allowing for the representation of magnitude and direction, which is essential for wave functions in quantum mechanics.
- 🌀 Complex numbers generalize the concept of positivity and negativity, enabling the representation of numbers that are neither purely positive nor negative, akin to points on the unit circle.
- 🌊 Complex numbers are particularly useful for describing waves, as they can capture both the amplitude and phase of oscillations.
- 📉 The real and imaginary parts of a complex number can represent different aspects of a wave, such as the up and down or left and right movements.
- 📚 Euler's formula, \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \), connects complex exponentials with trigonometric functions and is fundamental in understanding wave behavior.
- 🔍 Complex addition and multiplication are analogous to vector addition and the combination of amplitudes and phase shifts, respectively.
- 🌐 The multiplication of complex numbers by a unit length complex number with a varying phase angle can result in phase shifts of the original wave function.
- 📐 Fourier analysis uses complex numbers to construct arbitrary waveforms by adding sine and cosine waves, which is also applicable in quantum mechanics for superpositions.
- 🌌 In quantum mechanics, complex numbers are used to describe quantum states through superpositions of energy eigenfunctions, which can result in time-varying probability densities.
- 🔬 Quantum field theory utilizes plane waves as a basis to construct more complex quantum systems and is integral to the formulation of quantum electrodynamics.
Q & A
Why are complex numbers used in quantum mechanics instead of just real numbers?
-Complex numbers are used in quantum mechanics because they allow for the representation of two-dimensional phenomena, such as the amplitude and phase of waves, which are fundamental in describing quantum states and wave functions.
What is the defining feature of complex numbers according to the script?
-The defining feature of complex numbers is that they are two-dimensional, allowing for the representation of both magnitude and direction, which is essential for capturing the behavior of waves in quantum mechanics.
How does the script explain the concept of complex numbers in relation to real numbers?
-The script explains that complex numbers can be seen as a generalization of the positive and negative binary system of real numbers, where a complex number can exist on any number ray between positive and negative, not just on the two discrete options of the real number line.
Outlines
😲 The Complexity of Quantum Mechanics and Complex Numbers
The paragraph delves into the intricacies of quantum mechanics, highlighting the perplexing phenomena that challenge conventional understanding. It emphasizes the initial confusion surrounding the use of complex numbers in physics, questioning their necessity over real numbers. The explanation unfolds to reveal the two-dimensional nature of complex numbers, contrasting them with the one-dimensional real numbers. The author illustrates how complex numbers generalize the concept of positivity and negativity, allowing for a continuous range of directions in the complex plane. The analogy of waves is introduced to demonstrate how complex numbers can capture the essence of oscillating phenomena, such as sound or light waves, more effectively than real numbers.
📊 Euler's Formula and the Wave Representation of Complex Numbers
This section explores the representation of waves using complex numbers, focusing on Euler's formula as a pivotal tool in understanding wave dynamics. The paragraph explains how the real and imaginary parts of a complex number can be visualized as waves, with the real part representing the wave's position in the complex plane and the imaginary part indicating its phase shift. The importance of complex addition and multiplication in wave interference is introduced, setting the stage for a deeper exploration of how these mathematical operations relate to physical phenomena such as constructive and destructive interference.
🌀 The Role of Complex Numbers in Wave Interference and Fourier Analysis
The paragraph examines the application of complex numbers in wave interference and Fourier analysis. It discusses the process of adding complex numbers as akin to vector addition, leading to constructive or destructive interference patterns. The concept of multiplying complex numbers is also explored, showing how it affects the amplitude and phase of waves. The paragraph further illustrates how complex numbers can be used to generate arbitrary waveforms through the superposition of sine and cosine waves, drawing parallels with quantum mechanics and the use of basis functions in quantum systems.
🌌 Higher Dimensional Wave Functions and Complex Numbers in Quantum Mechanics
This section expands on the concept of complex numbers in higher-dimensional wave functions, using the example of a two-dimensional plane wave. It clarifies misconceptions about the physical representation of complex numbers and emphasizes their role in capturing the dynamics of waves in quantum mechanics. The paragraph also touches on the use of plane waves as a basis for constructing more complex waveforms in quantum systems.
Mindmap
Keywords
💡Quantum Mechanics
💡Complex Numbers
💡Wave Function
💡Amplitude
💡Phase
💡Euler's Formula
💡Fourier Analysis
💡Quantum Harmonic Oscillator
💡Probability Density
💡Superposition
💡Complex Conjugate
Highlights
Quantum mechanics involves complex numbers due to the two-dimensional nature of waves, challenging traditional real number systems.
Complex numbers are essential for capturing wave phenomena, such as the phase and amplitude changes in quantum systems.
The complex plane allows for a more nuanced representation of waves, beyond the simple positive and negative of real numbers.
Euler's formula, e^(iΘ) = cos(Θ) + i*sin(Θ), provides a bridge between complex numbers and wave functions in quantum mechanics.
Complex numbers are not just mathematical tools but are intrinsic to the physical description of quantum phenomena.
The concept of complex addition and multiplication is analogous to wave interference, with direct implications for quantum systems.
Complex numbers enable a unified approach to Fourier analysis, simplifying the representation of waveforms in quantum mechanics.
The quantum harmonic oscillator can be described using superpositions of energy eigenstates, demonstrating the utility of complex numbers.
Complex conjugates play a crucial role in quantum mechanics, particularly in calculating probability densities from wave functions.
The two-dimensional aspect of complex numbers does not correspond to physical space but to the directional properties of waves.
Plane waves are fundamental in quantum mechanics, serving as a basis for constructing more complex wave functions.
Complex numbers are not just an abstraction but have a tangible interpretation in terms of wave dynamics in quantum systems.
The video hints at a profound connection between local U(1) symmetry and electromagnetism in quantum electrodynamics, to be explored in future content.
Complex numbers are integral to understanding the quantum mechanical description of particles and their interactions.
The video emphasizes the importance of gaining intuition for complex numbers through practice and problem-solving in quantum mechanics.
The profound implications of complex numbers in quantum mechanics are only briefly touched upon, with deeper exploration promised for future videos.
Transcripts
quantum mechanics is a notoriously
complicated and confusing subject and
part of that is for good reason I mean
there really are these crazy phenomena
happening in the quantum worlds that are
challenging to imagine but uh one of the
reasons quantum mechanics is complicated
is that there are all these complex
numbers all over the place and at first
when you're getting into the subject
it's very confusing why are we using
complex numbers what do they mean why
can't we use real numbers isn't this
physics not math what why do we use
these crazy numbers right so that was
the source of confusion for me at least
for quite a long period of time and then
one day it clicked and I finally
understood and I was like oh okay yeah
actually that complex numbers are what
we want to use in quantum mechanics so
what are the complex numbers anyway
I think the most defining feature of the
complex numbers is that they're a
two-dimensional number and that seems
scandalous but if you've been using the
real number line then you're already
kind of complicit in using a
two-dimensional number system sort of
because when you write a real number you
write its magnitude but you also assign
it to one of the two number Rays either
positive or negative
so you already have this sense of
magnitude and direction in your number
system it's just that you only have two
options for the direction positive or
negative so the direction dimension of
the real numbers is just a discrete
binary thing rather than a continuous
thing
and all the complex numbers really are
is a generalization of that positive
negative binary that is the complex
numbers can be regarded as a
generalization of positivity and
negativity so that a number can be not
only on either the positive or negative
number Ray but also all of the number
rays in between
this is a very strange concept the first
time you see it
now if you're mostly used to using
numbers to count things this seems like
an affront to reason because after all
if you consider the number two for
example so you could have positive two
you could have negative two
but with the complex numbers you could
also have this two that's just somewhere
in this space of this somewhere in this
circle that has radius two the number
with magnitude 2 in the complex plane
can be at any one of these points
and that doesn't seem right because
you'll notice when two is up just purely
pointing straight up then it's neither
positive nor negative and yet it's still
two it still has the amplitude of two
uh that doesn't really fit into our
normal intuitions about counting right
it doesn't feel like it makes sense okay
but complex numbers are not about
counting in this kind of way
let's look at this
this is a wave what kind of wave I don't
know could be the surface of the ocean
it could be a sound wave where the
height represents the air pressure it
could be a wave of light just flopping
around in the electromagnetic field
whatever it is it's just a wave now this
is a very clean and pure wave that I'm
using to illustrate the point but it's a
wave nonetheless
so how can we use numbers to capture
what this wave is
well the first and most obvious thing is
if the wave is above the average level
if the wave is up we'll say it's a
positive number and if the wave is down
we'll say it's a negative number so
let's go ahead and color it in positive
and negative all right fair enough
that's not wrong
but let's pause time for a second
now look here
right at the point where the wave is
zero and going down
is that point really zero or does that
point exist in a harmonious Continuum
with the rest of the wave yeah it's zero
now but it's part of a bigger picture
and you know it's just gonna be changing
soon you know it'll soon be non-zero so
is that really zero in the same way that
a flat line is zero or does it somehow
have an amplitude even though it's also
kind of zero at the same time so do you
see an analogy between this thing that's
kind of zero and kind of not zero and
the example we were looking at earlier
when the two was pointing straight up
and it was also kind of zero and kind of
not zero now what's more look at this
point over here where it's also zero but
now it's going up all of the same
observations apply it's kind of zero but
it's not really it has some energy to it
even though it's zero it kind of is kind
of isn't so now we can see that this
number here is the opposite of the
previous number that we were looking at
because the previous one is on its way
down and this one is on its way up
let's transform our perspective and use
complex numbers to represent this wave
[Music]
this is what a complex wave looks like
notice that the amplitude is constant
and it's just the phase that's changing
so it's not moving up and down like the
other wave now this is a very pure wave
this is a wave of the form e to the I
Theta where in this case Theta is some
function of X and some function of time
now let's plot the real part of this
complex number that is how far left or
right the number is in the complex plane
and you'll see that we can recover that
wave we were looking at earlier
let's show the imaginary component of
this complex wave that is how far up and
down the wave is in the complex plane
and you'll notice here we get a wave
that looks very similar to the real part
but it's out of phase such that it takes
on maximum and minimum values when the
real part is at zero and vice versa by
the way the function e to the I Theta is
equal to cosine of theta plus I times
the sine of theta where I is the
imaginary unit this equation is known as
Euler's formula well one of his many
formulas and it gives us another way of
thinking about what this complex wave is
when you're first getting into complex
numbers you'll probably think about
Euler's formula as the definition of
what e to the I Theta is but as you
become more comfortable with e to the I
Theta you'll eventually just see that as
the wave and then the cosine and sine is
a way of splitting it up into the real
Parts in the imaginary part by the way
let me just quickly say on the topic of
the imaginary part imaginary numbers are
a misnomer okay they're just as real or
just as imaginary as the real number
members the complex numbers are
numerical structure they're a holistic
thing you know it doesn't make any sense
to say imaginary and real but whatever
this is the terminology you're stuck
with so it is what it is ultimately it's
a consequence of the fact that the
imaginary numbers were named before they
were understood and that's I think one
of Descartes greatest mistakes well that
in dualism but um anyway where were we
what are we talking about here to
understand why it's useful to be able to
represent a wave as a constant amplitude
complex number whose phase is changing
we'll have to take a look at the nature
of complex addition multiplication and
how this relates to wave interference
we'll do that in a moment but first I
want to make a quick comment about why
is it e to the I Theta gives us this
wave I don't have time in this video to
give a really satisfactory answer but I
can lead you in the right direction so
if you take derivatives of e to the I
Theta and and sine of theta and cosine
of theta you can write these functions
in terms of a Taylor series when you do
that you'll find that e to the I Theta
has a term of theta at every degree
whereas cosine has even terms and sine
has odd terms and if you look closely at
these series you can see that the terms
on the right hand side of the equation
zip together into the terms on the left
hand side of the equation so by taking
Taylor series you can prove to yourself
that in fact e to the I Theta is cosine
of theta plus I sine of theta
okay so that was a bit of a tangent but
I think it's important for you to know
now let's take a look at complex Edition
if we have any two complex numbers we
can add them just like their vectors so
we put them tail to tip or another way
of looking at it is the sum is the
diagonal of the parallelogram so here
I'm showing two complex numbers both of
which have magnitude 2 swinging around
in the complex plane
the complex number between them is their
sum and you'll see that since the
numbers both have magnitude 2 their sum
has a magnitude of anywhere from zero to
four zero when the two numbers are
perfectly out of phase four when the
numbers are perfectly in phase and some
intermediary value when the angles are
kind of in phase and kind of not in
phase and we'll see later how that has a
very close relationship to the idea of
constructive and destructive
interference in Waves by the way here's
the algebraic formula for complex
Edition and that's the same as adding
vectors like the animation shows
now let's let one of those twos become a
little bit longer and you can see a more
General representation of complex
addition and you still see this effect
where sometimes the numbers will align
with each other and will add it's the
magnitude sometimes they'll be
oppositely aligned and they'll sort of
destructively interfere so that's a
general phenomenon whenever you're
adding complex numbers
and we can also multiply any two complex
numbers so to multiply complex numbers
you multiply their amplitudes and you
add their phase angles relative to the
positive real number line so here I'm
showing a couple numbers both with
magnitude two they're swinging around in
the complex plane and I'm also showing
their product you'll notice that since
the two numbers both have magnitude 2
their product will always have magnitude
4 but the phase angle of their product
depends on the sum of the phase angles
of the individual twos and of course
that rule generalizes so any two complex
numbers to multiply them you multiply
their magnitudes and add to their phase
angles that's really useful because what
it means is that if we have a complex
number of unit length but some phase
angle in the plane we can multiply that
by some other complex number to shift
its phase by the unit 1 numbers phase
angle to demonstrate this idea of
rotating the phase of a complex number
by multiplying by a unit length complex
number consider the illustration that's
on your screen now here I have a blue
wave and that's the real part of the
function e to the i x so the classic
complex wave amplitude 1 function you
take the real part and it's basically
it's just cosine of x right now the wave
that's changing colors that's the same
function that's the real part of the
same function except now the function is
multiplied by some complex constant
let's call it a a has magnitude one but
its phase angle is changing so I'm
showing you here the blue line with the
dot at the end that's the number one in
the complex plane right the colorful
line with the dot at the end that's a
that's this unit length complex number
whose phase is swinging around and the
colorful wave that's changing color
that's the real part of the wave that
you get when you multiply by the complex
number a
now let's notice something when a is one
the two numbers overlap and the two
waves are the same
when a is negative one the two waves are
completely opposite so the sign of the
wave is switching at every moment what
was formerly up is now down and so on
but in all those angles in between the
waves are not just the same or not just
totally opposite but they're similar
their phase shifted by some amount
that's not a complete half wavelength
and so in this illustration you can see
how this notion of generalizing
positivity and negativity that we see in
the complex numbers actually has a
genuine natural a very real
interpretation we can see this even more
clearly if we put the sum of the two
waves into this illustration as well
now we can think about addition of waves
in two ways first you can sweep along
the x-axis and just add the value of the
two waves at any point and that gives
you the value of the third wave or you
can add the complex amplitudes of the
Waves you get a resulting complex number
that's the sum of those complex numbers
and then you multiply that by the way of
e to the IX and that gives you the sum
of the two waves so you see there's this
direct one-to-one relationship between
complex addition and the interference of
these waves
if you've studied signal processing then
you know that you can generate an
arbitrary waveform by adding sine waves
and cosine waves in the right amounts
and frequencies while complex numbers
let us create a more unified and
holistic way of doing Fourier analysis
by adding waveforms that are these e to
the i x kind of waves multiplied by a
complex coefficient and then summing
over frequencies and amplitudes in that
basis so the example I'm showing here of
generating a square wave you might
associate that more with like signal
processing or something in quantum
mechanics however we use the idea of
basis functions and superpositions all
the time consider for example the
quantum harmonic oscillator I already
made a video on the quantum harmonic
oscillator so I'm not going to rehash
all of the details here if you want to
see like the hamiltonian and Trojan's
equation and all that good stuff you can
check out that video but what I just
want to point out here is that we can
take this sum of energy eigenfunctions
the ground state and these few excited
States and if we add them all up we can
get the wave function of a particle
that's oscillating in the quantum
harmonic oscillator
and this is just one of the many ways
that these basis functions can be added
but when you look at it one thing to
notice is that if you just look at the
probability densities of each of the
energy eigenstates they're actually
stationary but when you add the
eigenstates because you're adding the
complex numbers and there's that complex
interference going on the subsequent
probability density of the sum of those
States the superposition of those States
is actually this thing that varies in
time
and so here we can see this cool kind of
Dynamics coming out of the Machinery of
complex numbers
my main point in this video is just to
get you familiar with the complex
numbers to show you that ultimately they
come from this notion of waves and we'll
see many examples of complex wave
functions going forward for example I'm
currently working on a video on the
hydrogen atom and so here I'll show you
just a little preview of that we have
the longitudinal component of the
hydrogen energy eigenstates so this is
what you get when you solve the
azimuthal equation you end up with the
hemholz differential equation and you
can derive the fact that hydrogen has
this quantized magnetic number M which
we learn about in chemistry from the
fact that the wave function has to loop
back in on itself as you go around
anyway we'll come back to this later in
the hydrogen video but for now I'd like
to look at a higher dimensional example
of a complex wave function
so for example here's a two-dimensional
plane Wave It's defined in the plane of
your screen it's constant amplitude and
the color represents the phase of the
wave function at every point
I've also superimposed these little
arrows and what the arrows represent is
numbers in the complex plane
now I want to use this to illustrate a
couple of points first when you look at
a picture like this it almost looks like
a vector field and there's a temptation
to think that the complex numbers are
embedded in this two-dimensional space
and that their direction is sort of
pointing in a direction in that space
this is a common misconception and I had
this misconception for a while when I
was learning quantum mechanics because
one of the things that confused me about
complex numbers was they're
two-dimensional right so why
I mean if you have a three-dimensional
wave function for example shouldn't you
have like some kind of three-dimensional
thing like how do you stick a
two-dimensional Arrow at a point in
space how does that even make sense but
I hope that based on everything you've
seen so far you realize that the
two-dimensionality of the complex
numbers actually is not about any
direction in physical space
the fact that the complex numbers are
two-dimensional is the fact that a wave
is up and down or left and right or back
and forth or high pressure low pressure
it's yin and yang when you see that then
the confusion goes away okay so that's
the first point the second point I want
to bring up when it comes to plane waves
is that these things you will see these
all over the place why a couple of
reasons one honestly it's kind of one of
the easiest solutions to all these wave
equations that you'll encounter in
quantum mechanics but also it can be
used as a Fourier basis to construct
these more complicated wave functions
like earlier when we looked at the
square wave and you could see how you
can make an arbitrary waveform by adding
a bunch of waves well if you take a
bunch of plane waves that satisfy for
example the Schrodinger equation or the
Klein Gordon equation or the Drac
equation although in that case you have
byspinner field it's more complicated
but whatever if you have plane waves
that satisfy some differential equation
and you add them you take their
superposition you can create these more
complicated systems that also satisfy
those equations and in fact in Quantum
field Theory the plane waves play a
essential constitutive role in the
second quantization that allows you to
actually make a Quantum field Theory
oh I almost forgot but earlier when we
were looking at the complex
multiplication let's go back to that
picture except now I'm showing you
something special so these are two
complex numbers that have amplitude 2
and their product has amplitude four but
notice this time the two numbers are
complex conjugates of one another that
means that the imaginary component has
flipped sine in other words it's been
mirrored about the real axis and so what
we're seeing here is a number of times
its complex conjugate and the result is
always stuck on the real number line why
is that well add the angles a complex
number and its complex conjugate always
have angles that add up such that you
get back on the real axis and for that
reason you'll often see the expression
PSI star PSI as a way of expressing the
amplitude squared of a complex number in
quantum mechanics PSI star and PSI are
very often the two slices of bread in an
operator sandwich but when you see PSI
star PSI without any operator in between
just think of that as amplitude squared
and by by the way if PSI is a wave
function then PSI star PSI is the
probability density relating to that
wave function so if you want to find
what's the probability of finding a
particle in some volume of space you
just integrate PSI star PSI over that
volume of space that gives you the
probability of finding the particle
there
okay I'm gonna end this video on a
cliffhanger by alluding to one of my
favorite ideas of all time which is
beautiful and profound and it relates to
the complex numbers and that is the idea
that in Quantum electrodynamics a local
U1 symmetry of the wave function implies
electromagnetism this is going to take a
while to unpack and I am going to come
back to this in a future video it's
probably going to take like an hour but
we're really going to get into it and
it's going to be awesome for now I'll
just show you these equations if you
know you know if you don't stay tuned
but it's a pretty profound concept
basically what it comes down to is that
when you're doing Quantum
electrodynamics you have a wave function
it's actually kind of four-way functions
in one but if you impose the condition
that you can swing your wave function
around in Phase space arbitrarily then
what you'll find is that in order to
keep the lagrangian density the same
that is in order to not affect the laws
of physics and Quantum electrodynamics
you need the gauge symmetry of the
electromagnetic 4 potential which is
what we actually see right so the four
potential is kind of like relativistic
voltage and there's this inherent tree
as a result of that which has to do with
the way that the derivatives not Bonds
the electrical Community Fields anyway
if there's a lot there I'm not going to
go into it now but just know that there
are really deep and profound ideas
relating to the complex numbers in
quantum mechanics this video today
really has only scratched the surface
but I hope it's given you at least some
intuition for the complex numbers
one final thing that I want to say
becoming familiar with the complex
numbers takes time it takes hours and
hours of like plotting equations and
solving problems and doing things so
you're not supposed to get the ideas
right away no one ever does but uh you
know hey a journey of a thousand miles
is you know one footstep at a time or
however that saying goes and I hope you
keep on going on that path because
physics is really one of the most
wholesome things a person can do I think
but maybe I'm biased all right well
that's the video hope you enjoyed it and
have a great day
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