Wave functions in quantum mechanics
Summary
TLDRThis educational video delves into the concept of wave functions in quantum mechanics, explaining their significance and connection to the position and momentum operators. It covers the derivation of wave functions, their role in the position and momentum representations, and the scalar product between states. The video also explores the transformation between position and momentum space through Fourier transforms, providing a foundational understanding of wave mechanics and its applications in 3D spatial dimensions.
Takeaways
- 🌊 Wave functions are a key concept in quantum mechanics, representing the position representation of quantum systems.
- 🎓 Students often first encounter quantum mechanics through wave functions, which are integral to understanding 3D spatial systems like potential barriers and the hydrogen atom.
- 📚 Wave functions are derived from the eigenvectors of the position operator, forming an orthonormal basis for state space.
- 🧮 The wave function \( \Psi(x) \) is defined as the expansion coefficient of a state vector in the position basis, essential for studying quantum systems.
- 🔄 The script explains the transition from position to momentum representation, highlighting the use of Fourier transforms in quantum mechanics.
- 🔢 The scalar product between two states in position representation is derived, emphasizing the foundational role of wave functions in quantum mechanics.
- 📉 The normalization of a wave function is discussed, showing how it relates to the probability interpretation of quantum mechanics.
- 🔄 The overlap matrix between position and momentum representations is explored, leading to the derivation of the transformation matrix.
- 🔗 The script connects the position and momentum wave functions through the Fourier transform, a fundamental result in wave mechanics.
- 📚 The generalization to three dimensions is briefly touched upon, showing how the principles apply to vector operators in quantum mechanics.
Q & A
What is a wave function in quantum mechanics?
-A wave function in quantum mechanics is a mathematical description of the quantum state of a particle or system of particles. It is used to determine the probability distribution of a particle's position, momentum, and other properties.
Why are wave functions important in quantum mechanics?
-Wave functions are important because they provide a way to calculate the probability of finding a particle in a particular location or state. They are fundamental to understanding quantum phenomena and are used in many quantum mechanical calculations.
What is the position representation in quantum mechanics?
-The position representation is a way of expressing quantum states using wave functions that describe the probability distribution of a particle's position in space. It is one of the possible representations in quantum mechanics, leading to the formulation known as wave mechanics.
How are wave functions related to the position and momentum operators?
-Wave functions are related to the position and momentum operators through eigenvalue equations. The position operator acting on a wave function gives the eigenvalue corresponding to a specific position, and similarly, the momentum operator acting on a wave function gives the eigenvalue corresponding to a specific momentum.
What is the significance of the commutator between position and momentum operators?
-The commutator between the position (X) and momentum (P) operators, given by [X, P] = iħ, is significant because it represents the fundamental uncertainty principle in quantum mechanics. It shows that position and momentum cannot be precisely known at the same time.
How do you derive the scalar product between two wave functions?
-The scalar product between two wave functions can be derived by expanding the state vectors in the position basis and using the orthonormality of the position eigenstates. The result is an integral over space of the product of one wave function and the complex conjugate of the other.
What is the normalization of a wave function and why is it important?
-The normalization of a wave function is the process of ensuring that the integral of the absolute square of the wave function over all space is equal to 1. It is important because it ensures that the probabilities derived from the wave function sum to 1, which is a requirement for a valid probability distribution.
How can one transform from the position representation to the momentum representation?
-One can transform from the position representation to the momentum representation using the overlap matrix, which is the integral of the position eigenstate times the momentum eigenstate. This leads to a Fourier transform relationship between the position space wave function and the momentum space wave function.
What is the role of the translation operator in quantum mechanics?
-The translation operator in quantum mechanics is used to move a quantum state by a certain amount in space. It is used in the calculation of the overlap matrix between position and momentum eigenstates and is crucial for deriving the relationship between position and momentum representations.
How do wave functions help in understanding 3D quantum systems?
-Wave functions help in understanding 3D quantum systems by providing a framework to describe the quantum states in three-dimensional space. They allow for the calculation of probabilities and the study of phenomena such as quantum tunneling and the behavior of particles in potential wells or barriers.
Outlines
🌌 Introduction to Quantum Mechanics and Wave Functions
The video begins with an introduction to quantum mechanics, focusing on wave functions. It explains that wave functions are a way to view quantum systems, particularly in the context of 3D spatial dimensions. The paragraph highlights the importance of wave functions in studying quantum mechanics due to their relevance in understanding systems like potential barriers, potential wells, and the hydrogen atom. The narrator outlines the plan for the video, which includes learning about wave functions as the position representation of state vectors, deriving known results from wave mechanics, and relating wave functions in real space to those in momentum space. The discussion starts with the position operator and momentum operator, their commutator, and how wave functions are related to these operators and their eigenstates.
📚 Deriving Scalar Product and Normalization from Wave Functions
This paragraph delves into the mathematical aspects of wave functions, starting with the eigenvalue equation for the position operator. It discusses the hermitian nature of the position operator and its orthonormal eigenstates. The concept of expanding any state vector in terms of this basis is introduced, leading to the definition of wave functions as the expansion coefficients in the position basis. The paragraph then moves on to discuss the scalar product between two states in the position representation, deriving the familiar expression from quantum mechanics. It also covers the normalization of a wave function, presenting the integral form that equates to the absolute square of the wave function.
🔄 Transition from Position to Momentum Representation
The focus shifts to transitioning from the position representation to the momentum representation. The paragraph introduces the overlap matrix, which is the bracket between position and momentum eigenstates. It encourages viewers to review the translation operator and its properties before proceeding. The translation operator's action on position eigenstates is discussed, leading to the calculation of the overlap matrix. The paragraph concludes with the derivation of a first-order differential equation for the bracket between position and momentum, setting the stage for solving it in the next part of the video.
🌐 Generalizing to Three Dimensions and Fourier Transforms
The final paragraph generalizes the concepts from one dimension to three dimensions, introducing the vector operators for position and momentum. It discusses the canonical commutation relations and how they relate to the eigenvalue equations for these operators. The paragraph also covers the orthonormality of the eigenstates and how an arbitrary state can be expanded in terms of these basis states. The video concludes with the derivation of the Fourier transform relationship between position and momentum wave functions, highlighting the integral involving a plane wave that connects these representations. The narrator summarizes the key takeaways from the video, emphasizing the foundational role of wave functions in quantum mechanics and their utility in studying 3D spatial problems.
Mindmap
Keywords
💡Wave Function
💡Position Representation
💡Momentum Representation
💡Eigenvalue Equation
💡Hermitian Operator
💡Scalar Product
💡Normalization
💡Fourier Transform
💡Translation Operator
💡Commutator
Highlights
Wave functions are a representation of quantum systems in the position basis.
Wave functions are fundamental in quantum mechanics and lead to wave mechanics.
Students often first encounter quantum mechanics through wave functions.
Wave functions are particularly useful for studying 3D systems like potential barriers and the hydrogen atom.
Wave functions are the expansion coefficients of state vectors in the position basis.
Position and momentum operators, X and P, are key to understanding wave functions.
Eigen kets of the position operator form an orthonormal basis for state space.
Any state vector can be expanded in the position basis using integration over position eigen kets.
The wave function in momentum space is defined similarly to the position space wave function.
Eigen kets of the momentum operator also form an orthonormal basis.
The scalar product between two states in position space is derived from the wave functions.
Normalization of a wave function is integral over position space of the absolute value squared of the wave function.
The transformation from position to momentum representation involves the overlap matrix.
The translation operator plays a crucial role in calculating the overlap matrix.
The overlap matrix between position and momentum eigen kets is proportional to e to the IPX over H bar.
The proportionality constant for the overlap matrix is determined to be 1 over the square root of 2 pi H bar.
Fourier transforms are used to relate wave functions in position and momentum representations.
Wave functions are a special case of representation in quantum mechanics, derived from state vectors and state space.
The video concludes with a generalization of the results to three dimensions, involving vector operators R and P.
Transcripts
hi everyone there is Professor am the
science many people who have only heard
about quantum mechanics in passing have
nonetheless heard about wave functions
but what is a wave function a wave
function is only one of many possible
ways of looking at a quantum system is
the so called position representation of
quantum mechanics and it leads to the
famous wave mechanics so why are wave
functions famous well there are two main
reasons why wave functions are famous
the first one is because many students
first learn about quantum mechanics in
terms of wave functions the second one
is because the language of wave
functions is most useful when studying
systems in 3d spatial dimensions these
include famous examples suggest
potential barriers potential wells all
the way to the hydrogen atom in this
video we will first of all learn that
wave functions are the position
representation of state vectors second
we will derive some known results from
wave mechanics such as the scalar
product between wave functions and third
will relate wave functions in real space
to wave functions in momentum space
let's get started to study wave
functions we need to start by looking at
the position operator X and the momentum
operator P and their commutator IH bar
this is because wave functions are
intimately related with these two
operators and their eigen ket's so let's
start by looking at the eigenvalue
equation for the position operator X hat
acting on the cat X gives us the
eigenvalue x acting on the cat x and to
be clear what I mean here by X hat is
the position operator by X I'm in the
position eigen value which is a quantity
that has unit of length and what I mean
by the cat X is the ket associated with
the eigenvalue x the position operator
is a hermitian operator and therefore
its eigen ket's form a basis that we can
choose to be orthonormal what that means
is that we can write the bracket between
X and X prime as equal to the Delta
function at X minus X prime once we have
defined the basis we can expand any ket
in our state space in terms of this
basis so let's do that let's pick a cat
sy and we expand it in the position
basis that we write it as an integral
over DX the expansion coefficient is the
bracket between X and soy and then
each of these is multiplied by the basis
cat X we are now ready to make one of
the most important definitions in the
whole of quantum mechanics we define the
wave function Phi of X as the expansion
coefficients of a cat side in the basis
formed by the eigen ket's of the
position operator wave functions are
fundamental in our study of quantum
mechanics and they form the formulation
known as wave mechanics my guess is that
most of you will have first learned
about quantum mechanics in terms of wave
functions what we can say here is that
wave functions are nothing more than a
special case of a representation they
are the representation of a state vector
psi corresponding to the eigen ket's of
the position operator so now we're able
to see how wave functions fit in the
more fundamental formulation of quantum
mechanics in terms of state vectors and
state space we can do something very
similar to what we have done for the
position representation in terms of the
momentum representation so we can look
at the eigenvalue equation for the
momentum operator P hat acting on the
ket P is equal to the eigenvalue p
acting on the ket P as before P hat
means the momentum operator P means the
momentum eigen value which has units of
mass times velocity and the ket P is the
ket associated with the eigen value P
the eigen ket's of P also form a basis
whose orthonormality condition is such
that the bracket between p and p prime
is equal to the Delta function of p
minus p prime we can expand an arbitrary
cat's-eye in the Pearson tation as an
integral over DP of the expansion
coefficient which is the bracket of p
with side times the basis ket P and we
can now define the wave function in
momentum space as sy bar of P as equal
to the expansion coefficient that is the
projection of the cat's-eye on the basis
P what we have learned so far in this
video are some of the most important
ideas in the whole of quantum mechanics
so do make sure that you understand
everything that we have discussed to
recap we have introduced the idea of
wave functions the wave function in the
position representation sigh of X is the
projection of a state vector psi onto
the position representation and the wave
function in the
mentum representation sidebar of P is
similarly the projection of a cat's eye
onto the basis P now that we have
introduced wave functions we are ready
to look at a number of operations in
state space in terms of wave functions
and to do that we will use the position
representation wave function Phi of X
the first property I want to discuss is
the scalar product between two states so
let's consider a first state psi in the
position representation integral over
the X bracket of X with side X and the
second state Phi in the position
representation as well integral over DX
prime bracket of X prime with Phi X
prime we can now calculate the scalar
product between these two states as the
bracket between sy and Phi and plugging
in the expansions in the position
representation we obtain integral over
DX sy X X integral over the X prime X
prime Phi X Prime rearranging this
expression we get integral with the ex
integral over the X Prime sy X X prime
Phi xx prime we recognize sy X as the
definition of the wave function star of
X X prime Phi as the wave function Phi
of X and xx prime as the Delta function
X minus X prime the Delta function makes
the integral of a DX prime very easy and
therefore we can write the scalar
product between psi and Phi as equal to
the integral over DX of Phi star X Phi X
those of you familiar with wave
mechanics will immediately recognize
this expression as the usual scalar
product between two wave functions but I
want to emphasize that in our case we
have derived it from more general ideas
based on scalar products between cats in
state space an easy next step we can
take now is to look at the normalization
of a wave function to do that we can see
that the norm of a cat which is the
scalar product of a cat sy with itself
and by using the formula we have just
the right we can write this down as the
integral over DX of side Star X Phi of X
and then we can rewrite this as the
integral of a DX of the absolute value
squared of Phi of X again this should be
formula very
Mille to those of you who know wave
mechanics so far we have looked at wave
functions both in the position and
momentum representations something that
is very useful when we do quantum
mechanics is to be able to go from one
representation to another so what I want
to do next is to look at how we can go
from the position representation to the
momentum representation to do that we
need to consider the overlap matrix
which in this case is the bracket
between X and P and if you need a
refresher about transforming from one
basis to another and about overlap
matrices take a look at the video linked
in the description before we try to
figure out what the overlap between X
and P is a word of encouragement it is a
bit tricky to get there but we will get
there so do bear with me
the first step we need to take is to
refresh our minds about the translation
operator there is a link to a full video
about the translation operator in the
description but let me just quote some
of the most important results that we
need for calculating the overlap matrix
the translation of a position eigen ket
by an amount alpha is given by a
translation operator T alpha which is
equal to e to the minus I alpha P over H
bar we can consider an infinitesimal
translation of minus epsilon to get T of
minus epsilon equals e to the I epsilon
P over H bar and we can Taylor expand
this exponential to obtain 1 plus I
Epsilon over H bar P plus a term of
order epsilon squared the action of the
translation operator T alpha on a ket X
is equal to another ket X plus alpha and
the corresponding expression in dual
space is such that the bra x times the
operator t alpha is equal to the bra X
minus alpha again if these results do
come at a surprise check the video link
in the description for more details now
that we have refreshed our mind by the
translation operator the first step we
need to take to calculate the overlap
between X and P is to look at the matrix
element of an infinitesimal translation
with respect to X and P so let's write X
T of minus epsilon P we then use the
results that we obtained here for the
dual space action of the translation
operator to rewrite this down as X plus
epsilon P we now copy the same matrix
element
x t- epsilon P but use instead the other
representation above in terms of the
Taylor expansion of the translation
operator to write X 1 plus I epsilon
over h-bar P plus a term of order
epsilon squared P putting this together
we obtain XP plus I epsilon over h-bar X
P hat P plus a term of order epsilon
squared one thing to note is that P hat
P is simply the eigen value P times the
ket P now that we have written down
these two expressions for the matrix
element of an infinitesimal translation
we are ready to set them equal to each
other and we can therefore rearrange
that equation to obtain eigen value P
bracket of X and P equals minus IH bar
and then we take the limit of epsilon
going to 0 of the rest of the terms that
depend on epsilon so we get the bracket
of X plus epsilon P minus the bracket of
X with P over epsilon the limit we have
written down is the definition of a
derivative and therefore we can write
this whole expression as equal to minus
IH bar the derivative with respect to X
of the bracket of X and P looking at the
left and at the right hand side of this
equation we see that we now have a first
order differential equation for the
bracket between X and P that we need in
our quest to transform from the position
to the momentum representations we can
solve this first order differential
equation by separation of variables so
we write d bracket of X with P over
bracket of X with P equals I over H bar
P DX we integrate both sides to obtain
the logarithm of the bracket between X
and P equals I over H power P X plus the
integration constant C and then we can
exponentiate both sides of the equation
to obtain that the bracket between X and
P is equal to n e to the IPX over H bar
the N in this expression is simply the
transformation of the integration
constant C where we exponentiated both
sides of the equation after quite a few
steps we have finally reached the
conclusion that the overlap matrix
between X and P is proportional to e to
the IPX over H bar and all we have left
to do now is to determine the
proportionality constant and let's start
with a fresh page
and we write down the final result we
obtained which is that the bracket
between X and P is equal to n times e to
the IPX over H bar to find the constant
n we'll start by looking at the
orthonormality condition in the X
representation the bracket of X with X
prime is equal to the Delta function of
X minus X prime and then we're going to
rewrite this bracket in the following
manner X 1 X prime equals x now we
insert the resolution of the identity in
the p representation so we write
integral of the DP of PP x prime and
operating through this gives us the
integral over DP of X P px prime the two
terms under the integral sign are simply
equal to the term above and it's complex
conjugate and we can therefore write the
whole thing as equal to absolute value
square root of n integral over DP of e
to the IPX minus X prime over H bar at
this point we must use one of the many
definitions of the Delta function which
tells us a delta function of X minus X
prime is equal to 1 over 2 pi integral
over D U of e to the IU X minus X Prime
and looking at our expression we see
that we have exactly this under the
integral sign if we make the
substitution u equals P over H bar we
can therefore evaluate the integral we
have to obtain M Squared 2 pi H bar
Delta function of X minus X Prime this
term we just obtained must be equal to
the Delta function above because both of
them are equal to the bracket between X
and X Prime we can therefore set them
equal to each other
and we obtain that the absolute value
squared of n is equal to 1 over 2 pi H
bar which in turn tells us that n is
equal to 1 over square root of 2 pi H
bar putting everything together we
finally can write the expression for the
overlap matrix between X and P which is
equal to 1 over square root of 2 pi H
bar times e to the IPX over H bar okay
so as I said at the beginning it would
take us a while to get here but finally
we have determined the transformation
matrix XP that allows us to go between
the position representation and the
momentum represented
so let's finish the job and let's write
sidebar of P equals the projection of
sion P which is equal to p1 sy which is
equal to P then we insert the resolution
of the identity in terms of X which is
integral of DX X X sine and then we
reexpress this whole thing as integral
over DX px X sine we can now collect the
fruit of our labor and we identify the
PX term as simply the conjugate of the
XP term above we also identify exercise
as simply the wave function Phi of X and
we can therefore write the whole thing
as 1 over square root of 2 pi H bar
integral over the X e to the minus IPX
over H bar Phi of X you should repeat
the same exercise for the converse
transformation but I'm just going to
write down the result which is that the
wave function of Phi of X which is equal
to the bracket between X and PSI is
equal to 1 over square root of 2 pi H
bar integral over DP of e to the IPX
over H bar sy bar P we are finally done
and we can say now that in order to go
from the wave function in the position
representation to the wave function in
the momentum representation or vice
versa
all we have to do is we have to
calculate the corresponding Fourier
transforms this is another general
result that will be familiar to those of
you who know about wave mechanics and
again we have reached this result by
using the more fundamental formalism of
state space and state vectors before we
conclude I very quickly want to
generalize the result we have obtained
in one I mentioned in terms of X and P
to three dimensions in terms of the
vectors R and P in three dimensions the
position operator R is a vector operator
made of three components which are X 1 X
2 and X 3 or equivalently X Y Z
similarly the momentum operator is also
a vector operator P and again it has
three components P 1 P 2 and P 3 or also
px py PZ putting these together we find
the canonical commutation relations
between these vector operators as
xj p k equals IH bar delta JK we can now
start looking at all the results we have
the right thing one i mentioned so the
eigenvalue equation for the position
operator is the operator are acting on
the ket R which gives you the eigen
value are acting on the cat are the
eigen ket's form a basis which is
orthonormal so the bracket between r and
r prime is equal to the Delta function
of R minus R prime and we can expand an
arbitrary cat's-eye in terms of the
position basis as the integral over the
r sy of our our website of r is the 3
dimensional wave function which is given
by the projection of the state psi on
the basis function r it works in exactly
the same way for the momentum operator
so p hat
acting on the ket P is equal to the
eigen value P acting on the ket P the
orthonormality relation reads bracket PP
prime equals Delta function of P minus P
prime we can expand the state sy in the
P basis as integral over DP sidebar of P
P and sy bar of T is the 3-dimensional
momentum wave function which is given by
the bracket between P and sy just like
in one dimension we can relate the
position and momentum wave functions
through the Fourier transform sy of R is
equal to 1 over 2 pi H bar to the power
of 3 hulls integral over DP e to the I P
dot R over H bar sidebar of P so let's
recap what we have accomplished in this
video we have started by looking at the
position operator X and the momentum
operator P whose commutator is IH bar
and we have defined representations
associated with the eigen ket's X and P
of these two operators these two
representations are very important in
quantum mechanics because they lead to
the formulation known as wave mechanics
which is the formulation by which most
students are first introduced to the
quantum world what we have seen is how
this formulation fits into the more
fundamental formulation based on state
space and we have identified the wave
function Phi of X as the representation
of a state side in the X faces
and the momentum space wave function
sidebar of P as the representation of
state sigh in the momentum basis having
defined wave functions from the
fundamental state space formulation of
quantum mechanics we can then derive the
usual results from wave mechanics such
as the scalar product between two states
PI and Phi which is the integral of a DX
of size star of X Phi of X and the
normalization of a cat
sy sy which is equal to the integral
over the X of the absolute value squared
of psi of X we have also been able to
relate the position representation to
the momentum representation and we have
found that the wave function psi of X is
related to the momentum space wave
function cipher of P through an integral
over a plane wave e to the IPX over H
bar which tells us that the momentum and
position wave functions are related to
each other via Fourier transform
in this video we have learned that wave
functions are simply a particular
representation of state vectors and we
have seen how they fit in the why the
formalism of quantum mechanics so what
next wave functions are extremely useful
in studying problems in 3d spatial
dimensions so we can now learn about
things like quantum tunneling of
particles all the way to the hydrogen
atom if you liked this video or if you
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