Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Eigenvalues and eigenfunctions Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

The schedule…

Part I Introduction: The Schrödinger equation and fundamental quantum systems Part II The formalism Part III Quantum mechanics of atoms and solids

Exam I Part I Exam II Part II + the hydrogen atom Final exam All material covered in the course

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Last time… 퐴 퐵

vectors?

퐴 + 퐵

vector spaces obey a simple set of rules Examples:

the polynomials of degree 2 the even functions all possible sound waves the complex numbers arithmetic progressions the solutions of the Schrödinger equation Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Last time…

The solutions of the Schrödinger equation (the ‘wave functions’) span a vector space

a Hilbert space is a vector space with a norm, and it is ‘complete’(large enough).

... much larger than Hilbert’s Grand Hotel 3 (e. g. ℝ, ℝ , 푃∞, 푓 )

ℕ, ℤ, and ℚ are ‘equally large’, but ℝ is larger (much larger!)

transcendental numbers are not lonely Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Today: Operators

What are operators? Observables? Hermitian operators? Determinate states?

What is a degenerate spectrum? Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

a linear transformation:

a map 푇 between two vector spaces 푉, 푊 푇: 푉 푊 such that

a) 푇(푣1 + 푣2) = 푇(푣1) + 푇(푣2) b) 푇(훼푣2) = 훼푇(푣2) Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

a linear transformation:

a map 푇 between two vector spaces 푉, 푊 푇: 푉 푊 such that

a) 푇(푣1 + 푣2) = 푇(푣1) + 푇(푣2) b) 푇(훼푣2) = 훼푇(푣2) 푾 푽 퐴 퐵 푇 푇(퐵)

푇(퐴 )

푇(퐴 ) + T(퐵) = 푇(퐴 + 퐵) 퐴 + 퐵 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

a linear transformation:

a map 푇 between two vector spaces 푉, 푊 푇: 푉 푊 such that

a) 푇(푣1 + 푣2) = 푇(푣1) + 푇(푣2) b) 푇(훼푣2) = 훼푇(푣2) 푾 푽

퐴 푇 훼푇 퐴 = 푇 훼퐴 푇(퐴 ) 훼퐴 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

a linear transformation:

a map 푇 between two vector spaces 푉, 푊 푇: 푉 푊 such that

a) 푇(푣1 + 푣2) = 푇(푣1) + 푇(푣2) b) 푇(훼푣2) = 훼푇(푣2) 푽 푽 퐻 Ψ = 퐸Ψ

Ψ 퐻 퐸Ψ Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

a linear transformation:

a map 푇 between two vector spaces 푉, 푊 푇: 푉 푊 such that

a) 푇(푣1 + 푣2) = 푇(푣1) + 푇(푣2) b) 푇(훼푣2) = 훼푇(푣2) 푽 푽

휓푛 푎 + 푛 + 1 휓푛+1

푎 +휓푛 = 푛 + 1 휓푛+1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

a linear transformation:

a map 푇 between two vector spaces 푉, 푊 푇: 푉 푊 such that

a) 푇(푣1 + 푣2) = 푇(푣1) + 푇(푣2) b) 푇(훼푣2) = 훼푇(푣2)

퐻 Ψ = 퐸Ψ

푎 +휓푛 = 푛 + 1 휓푛+1 other operators: 푥 푝 In Quantum Mechanics [푥 , 푝 ] Observables are represented by linear Hermitian operators Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

In Quantum Mechanics:

Observables are represented by linear Hermitian operators

What is an observable?

Who is observing?

What do you need to satisfy to be an observer? Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

In Quantum Mechanics:

Observables are represented by linear Hermitian operators

What does ‘Hermitian’ imply?

∗ 퐴 is Hermitian 퐴 is real 퐴 = 퐴

∗ ∗ ∗ 퐴 = Ψ∗퐴 Ψ d푥 퐴 = Ψ∗퐴 Ψ d푥 = 퐴 Ψ Ψ d푥

Ψ|퐴 Ψ 퐴 Ψ|Ψ Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

In Quantum Mechanics:

Observables are represented by linear Hermitian operators

In a finite dimensional vector space: operators can be represented as a matrix – with respect to a certain basis:

퐴푖푗 = 푒푖|퐴 |푒푗

(so the form of the matrix depends on the choice of basis) Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

In Quantum Mechanics:

Observables are represented by linear Hermitian operators

Determinate states return the same value 푞 after each measurement 푄

(e.g. 퐻 Ψ = 퐸 Ψ )

“(corresponding) Eigenvalue” “Eigenfunction of the Hamiltonian” For determinate states 휎 = 0

If two eigenfunctions have the same eigenvalue, we say that “the spectrum is degenerate” Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

퐻 Ψ = 퐸Ψ

퐸Ψ

Ψ

퐻 does not change the ‘direction’ of its eigenvectors (it does not change the state) Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that mirrors vectors in the 푥-푦 plane in the 푥-axis

푦

푥 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that mirrors vectors in the 푥-푦 plane in the 푥-axis

푦

푣 2

푣 1 푥

휆1 = 1

휆2 = −1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that mirrors vectors in the 푥-푦 plane in the 푦-axis

푦

푥 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that mirrors vectors in the 푥-푦 plane in the 푦-axis

푦

푣 2

푣 1 푥

휆1 = −1

휆2 = 1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that projects vectors in ℝ3 onto the 푥-푦 plane

푧

푦 푥 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that projects vectors in ℝ3 onto the 푥-푦 plane

푧

푣 2

푣 푦 푣 3 1 푥

휆1 = 0

휆2 = 휆3 = 1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that projects vectors in ℝ3 onto the 푥-푦 plane

푧

푣 2

푣 푦 푣 3 1 푥

휆1 = 0

(all the vectors in the 푥-푦 plane) 휆2 = 휆3 = 1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that mirrors vectors in ℝ3 into the origin

푧

푦 푥 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that mirrors vectors in ℝ3 into the origin

푧

푣 2

푣 푦 푣 3 1 푥

휆1 = 휆2 = 휆3 = −1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question Find the eigenvectors of the following operators:

The operator 푂 that mirrors vectors in ℝ3 into the origin

푧

푣 2

푣 푦 푣 3 1 푥

(all the vectors in ℝ3) 휆1 = 휆2 = 휆3 = −1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

푧

푣 2

푣 푦 푣 3 1 푥

(all the vectors in ℝ3) 휆1 = 휆2 = 휆3 = −1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’)

푧

푣 2

푣 푦 푣 3 1 푥

(all the vectors in ℝ3) 휆1 = 휆2 = 휆3 = −1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’, ‘eigenfunctions’, ‘eigenkets’ …)

푧

푣 2

푣 푦 푣 3 1 푥

(all the vectors in ℝ3) 휆1 = 휆2 = 휆3 = −1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’, ‘eigenfunctions’, ‘eigenkets’ …) 퐻 Ψ = 퐸Ψ

푧

푣 2

푣 푦 푣 3 1 푥

(all the vectors in ℝ3) 휆1 = 휆2 = 휆3 = −1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’, ‘eigenfunctions’, ‘eigenkets’ …) 퐻 Ψ = 퐸Ψ

푧 푎 +휓푛 = 푛 + 1 휓푛+1

푣 2

푣 푦 푣 3 1 푥

(all the vectors in ℝ3) 휆1 = 휆2 = 휆3 = −1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’, ‘eigenfunctions’, ‘eigenkets’ …) 퐻 Ψ = 퐸Ψ

푧 푎 +휓푛 = 푛 + 1 휓푛+1

푣 2

not an eigenstate of 푎 +

푣 푦 푣 3 1 푥

(all the vectors in ℝ3) 휆1 = 휆2 = 휆3 = −1 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’, ‘eigenfunctions’, ‘eigenkets’ …)

How to find eigenvectors: Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’, ‘eigenfunctions’, ‘eigenkets’ …)

How to find eigenvectors:

(in finite dimensional vector space) – solve the characteristic equation

퐴푣 = 휆푣 det 퐴 − 휆퐼 = 0 Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’, ‘eigenfunctions’, ‘eigenkets’ …)

How to find eigenvectors:

(in finite dimensional vector space) – solve the characteristic equation

퐴푣 = 휆푣 det 퐴 − 휆퐼 = 0

(in high dimensional Hilbert space) – e.g. by solving a differential equation

퐻 Ψ = 퐸Ψ Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’, ‘eigenfunctions’, ‘eigenkets’ …)

How to find eigenvectors:

(in finite dimensional vector space) – solve the characteristic equation

퐴푣 = 휆푣 det 퐴 − 휆퐼 = 0

(in high dimensional Hilbert space) – e.g. by solving a differential equation

퐻 Ψ = 퐸Ψ if the spectrum is non-degenerate then the eigenfunctions are orthogonal Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Conclusion: An operator does not change the ‘direction’ of its eigenvector

In quantum mechanics: An operator does not change the state of its eigenvectors (‘eigenstates’, ‘eigenfunctions’, ‘eigenkets’ …)

How to find eigenvectors:

(in finite dimensional vector space) – solve the characteristic equation

퐴푣 = 휆푣 det 퐴 − 휆퐼 = 0

(in high dimensional Hilbert space) – e.g. by solving a differential equation

퐻 Ψ = 퐸Ψ if the spectrum is non-degenerate then the eigenfunctions are orthogonal if the spectrum is discrete, then the Ψ’s are normalizable if the spectrum is continuous, then the Ψ’s are not normalizable Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Homework due Thursday 9 March : Reading: Sections 3.3 Summarize section 3.3