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![Page 1: Lesson 13 Functions and Dirac notation - Stanford University · Bra-ket notation and expansions on basis sets When we write the function in this different form as a vector containing](https://reader030.fdocuments.us/reader030/viewer/2022020120/5b4a7f0f7f8b9aac238c3ca9/html5/thumbnails/1.jpg)
5.2 Functions and Dirac notation
Slides: Video 5.2.1 Introduction to functions and Dirac notation
Text reference: Quantum Mechanics for Scientists and Engineers
Chapter 4 introduction
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Functions and Dirac notation
David MillerQuantum mechanics for scientists and engineers
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5.2 Functions and Dirac notation
Slides: Video 5.2.2 Functions as vectors
Text reference: Quantum Mechanics for Scientists and Engineers
Section 4.1 (up to “Dirac bra-ketnotation”)
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Functions and Dirac notation
Functions as vectors
Quantum mechanics for scientists and engineers David Miller
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Functions as vectors
One kind of list of arguments would be the list of all real numbers
which we could list in order as x1, x2, x3 … and so on
This is an infinitely long list and the adjacent values in the list
are infinitesimally close togetherbut we will regard these infinities as
details!
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Functions as vectors
If we presume that we know this list of possible arguments of the function
we can write out the function as the corresponding list of values, and
we choose to write this list as a column vector
1
2
3
f xf xf x
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Functions as vectors
For example we could specify the function at points spaced by some small amount x
with x2 = x1 + x, x3 = x2 + x and so on We would do this
for sufficiently many values of x and over a sufficient range of x
to get a sufficiently useful representation for some calculation
such as an integral
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Functions as vectors
The integral of could then be written as
Provided we choose x sufficiently smalland the corresponding vectors therefore sufficiently long
we can get an arbitrarily good approximation to the integral
2f x
1
2 21 2 3
3
f xf x
f x dx f x f x f x xf x
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Visualizing a function as a vector
Suppose the function is approximated by its values at three points
x1, x2, and x3and is represented as a vector
then we can visualize the function as a vector in ordinary geometrical space
f x
1
2
3
f xf xf x
f
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Visualizing a function as a vector
We could draw a vector whose components along three axes
were the values of the function at these three points
f
x1 axisf(x1)
f(x2)
f(x3) f
x1 axis
x2 axis
x3 axis
f(x1)f(x2)
f(x3)f(x)
f(x1)
f(x2)
f(x3)
x1 x2 x3
f(x)f(x1)
f(x2)
f(x3)
x1 x2 x3
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Visualizing a function as a vector
In quantum mechanics the functions are complex, not merely real
and there may be many elements in the vector possibly an infinite number
f
x1 axisf(x1)
f(x2)
f(x3) f
x1 axis
x2 axis
x3 axis
f(x1)f(x2)
f(x3)f(x)
f(x1)
f(x2)
f(x3)
x1 x2 x3
f(x)f(x1)
f(x2)
f(x3)
x1 x2 x3
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Visualizing a function as a vector
But we will still visualize the function and, more generally, the quantum mechanical state
as a vector in a space
f
x1 axisf(x1)
f(x2)
f(x3) f
x1 axis
x2 axis
x3 axis
f(x1)f(x2)
f(x3)f(x)
f(x1)
f(x2)
f(x3)
x1 x2 x3
f(x)f(x1)
f(x2)
f(x3)
x1 x2 x3
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5.2 Functions and Dirac notation
Slides: Video 5.2.3 Dirac notation
Text reference: Quantum Mechanics for Scientists and Engineers
Section 4.1 (first part of “Dirac bra-ket notation”)
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Functions and Dirac notation
Dirac notation
Quantum mechanics for scientists and engineers David Miller
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Dirac bra-ket notation
The first part of the Dirac “bra-ket” notation called a “ket”
refers to our column vector For the case of our function f(x)
one way to define the “ket” is
or the limit of this as
We put into the vector for normalization The function is still a vector list of numbers
f x
1
2
3
f x x
f x xf xf x x
0x
x
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Dirac bra-ket notation
We can similarly define the “bra” to refer a row vector
where we mean the limit of this as Note that, in our row vector
we take the complex conjugate of all the values Note that this “bra” refers to exactly the same
function as the “ket”These are different ways of writing the same function
f x
1 2 3f x f x x f x x f x x
0x
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Hermitian adjoint
The vectoris called, variously
the Hermitian adjoint the Hermitian transposethe Hermitian conjugatethe adjoint
of the vector
1 2 3a a a
1
2
3
aaa
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Hermitian adjoint
A common notation used to indicate the Hermitian adjoint
is to use the character “†” as a superscript†
1
21 2 3
3
aa
a a aa
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Hermitian adjoint
Forming the Hermitian adjoint is like reflecting about a -45º line
then taking the complex conjugate of all the elements
†1
2
3
aaa
1
2
3
aaa
1 2 3a a a = 1 2 3a a a
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Hermitian adjoint and bra-ket notation
The “bra” is the Hermitian adjoint of the “ket”and vice versa
The Hermitian adjoint of the Hermitian adjoint brings us back to where we started
††1 1
†2 21 2 3
3 3
a aa a
a a aa a
†f f †
f f
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Bra-ket notation for functions
Considering as a vectorand following our previous result
and adding bra-ket notation
where again the strict equality applies in the limit when
1
2 21 2 3
3
f x x
f x xf x dx f x x f x x f x xf x x
f x
n nn
f x x f x x
f x f x
0x
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Bra-ket notation for functions
Note that the use of the bra-ket notation hereeliminates the need to write an integral or a sum
The sum is implicit in the vector multiplication
n nn
f x x f x x
f x f x
1
2 21 2 3
3
f x x
f x xf x dx f x x f x x f x xf x x
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Bra-ket notation for functions
Note the shorthand for the vector product of the “bra” and “ket”
The middle vertical line is usually omittedthough it would not matter if it was still
there
g f g f
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Bra-ket notation for functions
This notation is also useful for integrals of two different functions
1
21 2 3
3
n nn
f x x
f x xg x f x dx g x x g x x g x xf x x
g x x f x x
g x f x
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Inner product
In general this kind of “product” is called an inner product in linear algebra
The geometric vector dot product is an inner product
The bra-ket “product” is an inner product
The “overlap integral” is an inner product
g f g f
g x f x dx
g f
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Inner product
It is “inner” because it takes two vectors and turns them into a number
a “smaller” entityIn the Dirac notation
the bra-ket gives an inner “feel” to this product
The special parentheses give a “closed” look
g f
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5.2 Functions and Dirac notation
Slides: Video 5.2.5 Using Dirac notation
Text reference: Quantum Mechanics for Scientists and Engineers
Section 4.1 (remainder of 4.1)
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Functions and Dirac notation
Using Dirac notation
Quantum mechanics for scientists and engineers David Miller
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Bra-ket notation and expansions on basis sets
Suppose the function is not represented directly as a set of values for each point in space
but is expanded in a complete orthonormal basis
We could also write the function as the “ket” (with possibly an infinite number of elements)
In this case, the “bra” version becomes
n x
n nn
f x c x
1
2
3
cc
f xc
1 2 3f x c c c
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Bra-ket notation and expansions on basis sets
When we write the function in this different form as a vector containing these expansion coefficients
we say we have changed its “representation”The function is still the same function
the vector is the same vector in our space We have just changed the axes we use to represent the
functionso the coordinates of the vector have changed
now they are the numbers c1, c2 , c3
f x f x
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Bra-ket notation and expansions on basis sets
Just as before, we could evaluate
so the answer is the same no matter how we write it
2
2
, ,
1
21 2 3
3
n n m mn m
n m n m n m nm nn m n m n
f x dx f x f x dx c x c x dx
c c x x dx c c c
cc
c c c f x f xc
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Bra-ket notation and expansions on basis sets
Similarly, with
we have
n nn
g x d x
1
21 2 3
3
cc
g x f x dx d d dc
g x f x
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Bra-ket expressions
Note that the result of a bra-ket expression like or
is simply a number (in general, complex) which is easy to see if we think of this as a vector
multiplicationNote that this number is not changed as we change the
representationjust as the dot product of two vectors
is independent of the coordinate system
f x f x g x f x
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Expansion coefficients
Evaluating the cn in
or the dn in
is simple because the functions are orthonormal Since is just a function
we can also write it as a ketTo evaluate the coefficient cm
we premultiply by the bra to get
n nn
f x c x n n
n
g x d x n x
n x
m
m n m n n mn mn n
x f x c x x c c
n
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Expansion coefficients
Using bra-ket notationwe can write as
Because cn is just a numberit can be moved about in the product
Multiplication of vectors and numbers is commutativeOften in using the bra-ket notation
we may drop arguments like xThen we can write
n nn
f x c x n n n n n n
n n n
f x c x x c x x f x
n n n n n nn n n
f c c f
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State vectors
In quantum mechanicswhere the function f represents the state of the quantum mechanical system
such as the wavefunctionthe set of numbers represented by the bra or
ket vector represents the state of the system
Hence we refer toas the “state vector” of the system
and as the (Hermitian) adjoint of the state vector
ff
ff
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State vectors
In quantum mechanics the bra or ket always represents either
the quantum mechanical state of the system
such as the spatial wavefunctionor some state the system could be in
such as one of the basis states
x
n x
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Convention for symbols in bra and ket vectors
The convention for what is inside the bra or ket is looseusually one deduces from the context what is meant
For example if it is obvious what basis we were working with
we might use to represent the nth basis function (or basis “state”)
rather than the notation or The symbols inside the bra or ket should be enough to
make it clear what state we are discussingOtherwise there are essentially no rules for the notation
n
n x n
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Convention for symbols in bra and ket vectors
For example, we could write
but since we likely already know we are discussing such a harmonic oscillator
it will save us time and space simply to writewith 0 representing the quantum number
Either would be correct mathematically
20.375
The state where the electron has the lowestpossible energy in a harmonic oscillator with
potential energy x
0
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