lesson 13 functions and dirac notation - | stanford … · for scientists and engineers chapter 4...

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

Functions and Dirac notation

David MillerQuantum mechanics for scientists and engineers


Slides: Video 5.2.2 Functions as vectors


Section 4.1 (up to “Dirac bra-ketnotation”)


Functions as vectors

Quantum mechanics for scientists and engineers David Miller


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!


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


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


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

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


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


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


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


Slides: Video 5.2.3 Dirac notation


Section 4.1 (first part of “Dirac bra-ket notation”)


Dirac notation


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

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

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

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

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

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

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


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


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


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

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

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


Slides: Video 5.2.5 Using Dirac notation


Section 4.1 (remainder of 4.1)


Using Dirac notation


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


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


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


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

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

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

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

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

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

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

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

lesson 13 functions and dirac notation - | stanford … · for scientists and engineers chapter 4...

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