3 Hilbert Space and Vector Spaces 3 A

3. Hilbert Space and Vector Spaces 3 A. 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

3 B. 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 f 1| and f 2| and any two complex numbers c 1 and c 2, 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

3 C. Coordinate Bases What’s a Coordinate Basis? • In 3 D 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 3 D of complete continuous bases: • It is also common to simultaneously use mixed continuous and discrete bases

3 D. 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

3 E. 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.

3 F. 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

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 • We therefore have:

Sample Problem If | is normalized and U is unitary, show that U | is normalized. • U | is normalized Sample Problem If H is Hermition, show that |H | is real • |H | is real

3 G. 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 S 2, acting on these states, yields Write S 2 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 3 H, 3 I 3. 3 Wednesday 4 A, 4 B, 4 C 3. 4, 3. 5 Friday 4 D, 4 E 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: Completeness • Note that each column of V is components of | ’i in the unprimed basis

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

3 H. 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 real • Eigenvectors of Hermitian operators are real: real • Eigenvectors of Unitary operators are complex numbers of magnitude one

Orthogonality of Eigenvectors • Hermitian operators can act equally well to the left or to the right on eigenvectors • Let |v 1 and |v 2 be eigenvectors of A with distinct eigenvalues • Then |v 1 and |v 2 will be orthogonal to each other The same property applies to Unitary operators U • Let |v 1 and |v 2 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 that has a trunk • Often, the eigenvalues correspond to physically measurable quantities • In particular, the Hamiltonian is often chosen as one of the observables

3 H. 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 S 2 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†S 2 V 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(S 2 - 1) = 0 • For each eigenvalue, solve S 2|v = |v • = 0: • = 2: • The eigenvalues of S 2 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†S 2 V is now diagonal

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