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:

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):

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

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

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:

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

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

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

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

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:

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:

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

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

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 *

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

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 |,

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

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

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 :

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

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

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:

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

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

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:

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

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

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

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

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

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

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:

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

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

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

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

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

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:

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

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

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

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

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

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

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

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

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

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

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