1. 3. Hilbert Space and Vector Spaces
3A. Hilbert Space
Real Vector Spaces
A real vector space is a set V with elements v with the following properties:
You can add them
You can multiply them by real numbers:
We define multiplication in either order the same:
The multiplication has associate and distributive properties:
We also define an inner product (dot product):
This dot product is linear in both arguments:
It is also positive definite:

2. Inner Product for Wave Functions
Consider wave functions at a specific time
Define the inner product of two wave functions:
This inner product is linear in its second argument and anti-linear in the first:
It is also positive definite:
Some other properties can be proven using just these two properties:
Proof complicated in general:
Schwartz inequality (homework):

3. Hilbert Space
We are especially interested in normalized wave functions
Hilbert Space, denoted, is the set of wave functions satisfying
Hilbert space is a complex vector space with a positive definite inner product
You can add wave functions:
You can multiply them by complex numbers:
Let’s prove closure under addition:
positive

4. Extended and Restricted Hilbert spaces
Hilbert space has only the restriction
Sometimes, we wish to consider wave functions that do not fit this restriction, like eikr
In this case, we are looking at a superspace of Hilbert space
Sometimes, we want to consider more restrictive constraints
|(r)| < 
(r) continuous
’(r) exists, ’’(r) exists …
In this case, we are looking at a subspace of Hilbert space
These mathematical niceties won’t concerns us much

5. 3B. Dirac Notation and Covectors
Vectors/Kets
A wave function and its Fourier transform contain equivalent information
There are other ways to denote a wave function as well:
We might want to consider cases not denoted just by a wave function
Spin
Multiple particles
Indefinite number of particles
To keep our notation as general as possible, we will denote this information as an abstract vector, and clarify it by drawing a ket symbol around it:

6. Covectors / Bras
A covector, such as f is a linear mapping from vectors to complex numbers
Linear means:
Dirac drew a bra around a covector to signal that it’s a covector
When you put a bra with a ket, you get a bracket
The extra bar is deleted for brevity
Some examples of covectors/bras:
The position bra r|
The momentum bra k|
For any |  , define | by

7. Covectors as a Vector Space
For any two covectors f1| and f2| and any two complex numbers c1 and c2, define the covector by
Covectors form a vector space * as well
In fact, there is a one-to one-correspondence between vectors and covectors
Mathematically, this means the two vector spaces are the same,  = *
The process of turning a bra into its corresponding ket and vice-versa is called Hermitian Conjugation and is denoted by a dagger

8. Sample Problem
What are the corresponding kets (wave functions) for the position and momentum bras?

9. Hermitian Conjugation and Complex Numbers
Consider the Hermitian Conjugate of a linear combination:
This is defined by how it acts on an arbitrary ket:
Sums remain sums under Hermitian conjugation
But Hermitian conjugation takes the complex conjugate of complex numbers

10. What’s a Coordinate Basis?
3C. Coordinate Bases
What’s a Coordinate Basis?
In 3D space, it is common to write vectors in terms of components:
In a complex vector space, we would similarly like to write
It is complete if every vector can be written this way
It is independent if there is no non-trivial way to write the zero ket this way
It is orthogonal if the inner product between any distinct pair vanishes
It is orthonormal if:

11. Making an Orthonormal Basis
Any complete basis can be made into a complete, independent, orthonormal basis
To make it independent, throw out redundant basis vectors:
To make it orthogonal, iteratively produce new vectors orthogonal to each other
To make it orthonormal, normalize them
Our basis is now:

12. Pulling out the Coefficients
In general, given | and a basis, we would like to find the complex numbers ci such that
Assuming the basis is complete, this can always be done, but actually doing it may be difficult
For an orthonormal basis, this is easy. Act on | with n|:
Dimensionality of a Vector Space
The dimensionality of a vector space is the number N of basis vectors in a complete, independent basis
Usually infinity for us

13. Continuous Bases
Sometimes, it is more useful to work with bases that are labeled by real numbers rather than discrete integers
Then a general ket can be written as
An orthonormal basis would have the property:
The coefficient functions are easily determined:
Some examples in 3D of complete continuous bases:
It is also common to simultaneously use mixed continuous and discrete bases

14. Operators and Linear Operators
3D. Operators
Operators and Linear Operators
An operator is anything that maps vectors to vectors:
Usually, omit the parentheses
I will (usually) denote operators by capital letters A
We will focus almost exclusively on linear operators:
When I say “operator”, linear is implied
Operators themselves form a vector space:
Technically, this space is *

15. Examples of Operators The identity operator 1:
I will rarely bold the 1
The position operators R = (X,Y,Z), which multiply by the coordinate:
The momentum operators: P = (Px,Py,Pz) which take a derivative:
The Hamiltonian operator:
The Parity operator , which reflects the wave function through the origin:
For any ket |1 and bra 2|, define |12| by

16. The completeness relation
Let {|i} be a complete orthonormal basis
Consider the following expression:
To figure out what this is, let it act on an arbitrary ket |
We already know:
Therefore:
Since this is true for all |,

17. Operators acting on covectors (bras)
We normally think of operators acting on vectors to produce vectors:
Operators can also act on covectors to produce covectors
Logically this could be written as A(|), but instead we write this as
This is defined by how we it acts on an arbitrary vector:
Pretty much ignore parentheses
We can just as easily define an operator by how it acts on covectors as on vectors

18. Multiplying Operators
The product of two operators A and B is defined by
From this property, you can prove the associative property for operators:
Bottom line: Ignore the parentheses:
You can also show the distributive law:
Operator multiplication is usually not commutative

19. Definition of the Commutator:
3E. Commutators
Definition of the Commutator:
Usually, two operators do not commute:
Define the commutator of two operators A and B by
Warning: avoid using [] as grouping when doing commutators
Let’s work out the commutator of X and Px
To do so, let X and Px act on an arbitrary wave function :

20. Some Simple Commutators:
Generalize:
Some other simple ones:
Some easy to prove identities to help you get more commutators:

21. Sample Problem
The angular momentum operators are defined as L = R  P. Find the commutator of Lz with all components of R.

22. 3F. Hermitian Adjoints of Operators
Definition of the Adjoint of an Operator:
Let A be an arbitrary operator. Define the Hermitian adjoint A† by
where |  = |†
This is a linear operator:
A very useful formula:
From this, easy to show:

23. Conjugates of products of operators
What is (AB)†?
General Rules for Finding Adjoints of Expressions:
For sums or differences, treat each term separately
For things that are multiplied (bras, kets, operators), reverse the order
Replace A by A† for each operator
Replace bras  kets, | |
Replace complex numbers c by their complex conjugates

24. Find the Hermitian adjoint of the equation:
Sample Problem
Find the Hermitian adjoint of the equation:
Each term in the sum is treated separately
Reverse the order of everything multiplied
Adjoint everything
Did we do it right?
Answer: both are correct
Warning: Don’t reverse the order of labels inside a ket

25. Hermitian and Unitary Operators
A Hermitian operator A is one such that
A Unitary operator U is one such that
Proving one of these is sufficient
Generally, easiest to prove by inserting an arbitrary ket and bra:
For Hermitian, equivalent to
Sufficient (and sometimes easier) to prove it with basis vectors:

26. Examples of Hermitian Operators:
The three position operators:
The three momentum operators:
The Hamiltonian

27. Examples of Unitary Operators
Define the translation operators by how they translate the position basis vectors:
Taking the Hermitian conjugate
We therefore have:
Define the rotation operators by how they rotate the position basis vectors:
Where  is a 33 matrix satisfying

28. Sample Problem Sample Problem
If | is normalized and U is unitary, show that U | is normalized.
U | is normalized
|H | is real
Sample Problem
If H is Hermition, show that |H | is real

29. Matrices for Vectors and Covectors
3G. Matrix Notation
Matrices for Vectors and Covectors
Suppose we have an orthonormal basis {|1, |2, |3, …}
It is common to write vectors |v by simply listing their components
Put the components vi together into a column matrix
A covector w| can also be written as a list of numbers
Put the components together into a row matrix:
The inner product is then simply matrix multiplication

30. Matrices for Operators
For an arbitrary operator A:
Write A then as a square matrix:
Any manipulation of operators, bras, and kets can be done with matrices

31. Why Matrix Notation?
Often easier to understand if we think of bras, kets, operators as concrete matrices, rather than abstract objects
Even though our matrices are usually infinite dimensional, sometimes we can work with just a finite subspace, making them finite
In such cases, computers are great at manipulating matrices

32. Sample Problem
For two spin- ½ particles, the spin states have four orthonormal basis states:
The operator S2, acting on these states, yields
Write S2 as a matrix in this basis.
First, pick an order for your basis states:

33. Hermitian Conjugates in Matrix Notation
For a vector and its corresponding covector,
So, if thought of as a matrix, | becomes | by changing the column into a row and complex conjugating
For an operator:
For complex numbers, Hermitian conjugation means complex conjugation
General rule: Turn rows  columns and complex conjugate

34. Announcements ASSIGNMENTS Day Read Homework Today 3H, 3I 3.3
Wednesday 4A, 4B, 4C 3.4, 3.5
Friday 4D, 4E 3.6
All homework returned
On the web:
Chapter 3 Slides
Chapter 2 Homework Solutions
9/15

35. Changing Bases (1)
All the components of a vector, covector, or operator, depend on basis {|i}
What if we decide to change to a new orthonormal basis {|’i}?
Define the conversion matrix:
This is a unitary matrix:
Note that each column of V is components of |’i in the unprimed basis
Completeness

36. Changing Bases (2)
We can now use V to convert a vector, covector, or operator
Vectors:
Covectors:
Operators:
Note: The primes don’t mean new objects, but the same old objects in a new basis

37. 3H. Eigenvectors and Eigenvalues Definition, and some properties
An eigenvector of an operator A is a non-zero vector |v such that
where a is a complex number called an eigenvalue
If |v is an eigenvector, so is c|v
We can normalize eigenvectors when we want to
Eigenvectors of Hermitian operators are real:
Eigenvectors of Unitary operators are complex numbers of magnitude one
real
real

38. Orthogonality of Eigenvectors
Hermitian operators can act equally well to the left or to the right on eigenvectors
Let |v1 and |v2 be eigenvectors of A with distinct eigenvalues
Then |v1 and |v2 will be orthogonal to each other
The same property applies to Unitary operators U
Let |v1 and |v2 be eigenvectors of U with distinct eigenvalues
Recall:
We see that
Consider the expression:

39. Observables and Basis Sets
Suppose that the eigenvectors of a Hermitian operator A form a complete set (mathematically, they span the space), so that we can always write:
This can always be done in finite dimensional space, often in infinite
A Hermitian operator with this property is called an observable
Suppose we start with a complete set of eigenstates {|n}
Throw out any redundant ones
Construct an orthogonal basis set by the usual procedure:
Note that states with different eigenvalues don’t mix, so still eigenstates
Now make them orthonormal by the usual procedure:
Still eigenstates!
We end up with an orthonormal basis set which are eigenstates of A

40. Labelling Basis States
When we choose a basis set that are eigenvalues of observable A, often label them by their eigenvalues
If we work in this bases, the matrix for A will be diagonal
Switching to this basis is called diagonalizing A
Let B be another observable that commutes with A, so
If you act on any eigenstate of A with B, it is still an eigenstate of A:
This implies that B will be block diagonal
When you find eigenstates of B, they will exist just within these subspaces with the same A eigenvalues
You can label them by both their eigenvalues under A and B

41. Complete Sets of Commuting Observables
We can continue this procedure, until eventually our states are uniquely labeled by their eigenvalues
The observables you are using are called a complete set of commuting observables
They must all commute with each other
Eigenstates can be labelled just by their eigenvalues
Why is this a good idea?
Oddly, making the requirements more stringent can make them easier to find
Think of an animal whose name ends in -ant
Often, the eigenvalues correspond to physically measurable quantities
In particular, the Hamiltonian is often chosen as one of the observables
that has a trunk

42. 3H. Finding Eigenvalues and Eigenvectors How to find Eigenvalues
Suppose an operator is given in matrix form
How can we find its eigenvectors and eigenvectors?
We want to solve:
Rewrite this as:
This is N equations in N unknowns
Generally has a unique solution: |v = 0
Unless det(A – 1) = 0 !
To find eigenvalues, solve det(A – 1) = 0
This is an N’th order polynomial
It has exactly N complex roots, though not necessarily distinct
For Hermitian matrices, these roots will be real

43. Procedure for Eigenvalues and Eigenvectors
To find the eigenvalues and orthonormal eigenvectors of operator A:
First find roots of det(A – 1) = 0
This produces N eigenvalues {n}
Then, for each eigenvalue n:
Solve the equation A|vn = n|vn
If you have any degenerate eigenvalues n, you may have to be careful to keep them orthogonal
You then need to normalize them: vn|vn = 1
To change basis to the new basis vectors
Make the basis transformation matrix V from the eigenvectors |vn
You can then use this to transform vectors, covectors or operators to the new basis
In particular, the operator A will be diagonal and will have the eigenvalues on the diagonal
You don’t need to calculate it

44. Shortcuts for Eigenvalues and Eigenvectors
If a matrix is already diagonal, the eigenvalues and eigenvectors are trivial
The eigenvalues are the values on the diagonal
The eigenvectors are the unit vectors in this basis
If a matrix is block diagonal, you can divide it into two (or more) smaller and easier problems
If there is a common factor in an operator, take it out
Matrix without c has identical normalized eigenvectors
Eigenvalues are identical except multiplied by c

45. Sample Problem (1) For the S2 matrix found before,
(1) Find the eigenvalues and orthonormal eigenvectors
(2) Find the matrix V that relates the new basis to the old, and demonstrate explicitly that V†S2V is now diagonal
(3) Write the state |2 = |+– in terms of the eigenvector basis.
Take the common factor of 2 out.
Note that it is block diagonal
Two of the eigenvectors and eigenvalues are now trivial
All that’s left is the middle

46. Sample Problem (2)
(1) Find the eigenvalues and orthonormal eigenvectors . . .
Solve the equation det(S2 - 1) = 0
For each eigenvalue, solve S2|v = |v
 = 0:
 = 2:
The eigenvalues of S2 will be 2 times these, or 0 and 22

47. Sample Problem (3)
(1) Find the eigenvalues and orthonormal eigenvectors
(2) Find the matrix V that relates the new basis to the old, . . .

48. Sample Problem (4)
. . .and demonstrate explicitly that V†S2V is now diagonal

49. Sample Problem (5)
(3) Write the state |2 = |+– in terms of the eigenvector basis.