Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

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

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

Last time… 𝐴 𝐵

𝐴 + 𝐵

vectors?

vector spaces obey a simple set of rules

the polynomials of degree 2

the even functions

all possible sound waves

the complex numbers

arithmetic progressions

the solutions of the Schrödinger equation

Examples:

Lecture 13: Eigenvalues and eigenfunctions

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

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

... much larger than Hilbert’s Grand Hotel

Introduction to Quantum Mechanics I

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

(e. g. ℝ, ℝ3, 𝑃∞, 𝑓 )

Last time…

transcendental numbers are not lonely

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

OperatorsToday:

What are operators?

Observables?

Hermitian operators?

Determinate states?

What is a degenerate spectrum?

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

a map 𝑇 between two vector spaces 𝑉, 𝑊𝑇: 𝑉 𝑊

such that

a) 𝑇(𝑣1 + 𝑣2) = 𝑇(𝑣1) + 𝑇(𝑣2)b) 𝑇(𝛼𝑣2) = 𝛼𝑇(𝑣2)

a linear transformation:

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

a map 𝑇 between two vector spaces 𝑉, 𝑊𝑇: 𝑉 𝑊

such that

a) 𝑇(𝑣1 + 𝑣2) = 𝑇(𝑣1) + 𝑇(𝑣2)b) 𝑇(𝛼𝑣2) = 𝛼𝑇(𝑣2)

𝐴𝐵

𝐴 + 𝐵𝑇( 𝐴) + T(𝐵) = 𝑇( 𝐴 + 𝐵)

𝑇(𝐵)

𝑇( 𝐴)

𝑽𝑾

𝑇

a linear transformation:

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

a linear transformation:

a map 𝑇 between two vector spaces 𝑉, 𝑊𝑇: 𝑉 𝑊

such that

a) 𝑇(𝑣1 + 𝑣2) = 𝑇(𝑣1) + 𝑇(𝑣2)b) 𝑇(𝛼𝑣2) = 𝛼𝑇(𝑣2)

𝐴

𝛼 𝐴𝑇( 𝐴)

𝑽𝑾

𝛼𝑇 𝐴 = 𝑇 𝛼 𝐴𝑇

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

a linear transformation:

a map 𝑇 between two vector spaces 𝑉, 𝑊𝑇: 𝑉 𝑊

such that

a) 𝑇(𝑣1 + 𝑣2) = 𝑇(𝑣1) + 𝑇(𝑣2)b) 𝑇(𝛼𝑣2) = 𝛼𝑇(𝑣2)

Ψ

𝑽𝑽

𝐸Ψ 𝐻

𝐻Ψ = 𝐸Ψ

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

a linear transformation:

a map 𝑇 between two vector spaces 𝑉, 𝑊𝑇: 𝑉 𝑊

such that

a) 𝑇(𝑣1 + 𝑣2) = 𝑇(𝑣1) + 𝑇(𝑣2)b) 𝑇(𝛼𝑣2) = 𝛼𝑇(𝑣2)

𝜓𝑛

𝑽𝑽

𝑛 + 1 𝜓𝑛+1

𝑎+

𝑎+𝜓𝑛 = 𝑛 + 1 𝜓𝑛+1

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

a linear transformation:

a map 𝑇 between two vector spaces 𝑉, 𝑊𝑇: 𝑉 𝑊

such that

a) 𝑇(𝑣1 + 𝑣2) = 𝑇(𝑣1) + 𝑇(𝑣2)b) 𝑇(𝛼𝑣2) = 𝛼𝑇(𝑣2)

𝐻Ψ = 𝐸Ψ

𝑎+𝜓𝑛 = 𝑛 + 1 𝜓𝑛+1

𝑥 𝑝

[ 𝑥, 𝑝]

other operators:

In Quantum Mechanics

Observables are represented by linear Hermitian operators

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

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?

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

In Quantum Mechanics:

Observables are represented by linear Hermitian operators

What does ‘Hermitian’ imply?

𝐴 is Hermitian 𝐴 is real 𝐴 = 𝐴∗

𝐴 = Ψ∗ 𝐴 Ψ d𝑥 𝐴∗

= Ψ∗ 𝐴 Ψ d𝑥

∗

= 𝐴Ψ∗Ψ d𝑥

Ψ| 𝐴Ψ 𝐴Ψ|Ψ

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

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:

𝐴𝑖𝑗 = 𝑒𝑖| 𝐴|𝑒𝑗

(so the form of the matrix depends on the choice of basis)

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

In Quantum Mechanics:

Observables are represented by linear Hermitian operators

Determinate states return the same value 𝑞 after each measurement 𝑄

(e.g. ) 𝐻Ψ = 𝐸Ψ

“Eigenfunction of the Hamiltonian”

“(corresponding) Eigenvalue”

If two eigenfunctions have the same eigenvalue,

we say that “the spectrum is degenerate”

For determinate states 𝜎 = 0

Lecture 13: Eigenvalues and eigenfunctions

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

𝐻Ψ = 𝐸Ψ

Ψ

𝐸Ψ

𝐻 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 𝑂 that mirrors vectors in the 𝑥-𝑦 plane in the 𝑥-axis

𝑥

𝑦

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question

Find the eigenvectors of the following operators:

The operator 𝑂 that mirrors vectors in the 𝑥-𝑦 plane in the 𝑥-axis

𝑥

𝑦

𝑣1

𝑣2

𝜆1 = 1

𝜆2 = −1

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question

Find the eigenvectors of the following operators:

The operator 𝑂 that mirrors vectors in the 𝑥-𝑦 plane in the 𝑦-axis

𝑥

𝑦

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question

Find the eigenvectors of the following operators:

The operator 𝑂 that mirrors vectors in the 𝑥-𝑦 plane in the 𝑦-axis

𝑥

𝑦

𝑣1

𝑣2

𝜆1 = −1

𝜆2 = 1

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question

Find the eigenvectors of the following operators:

The operator 𝑂 that projects vectors in ℝ3 onto the 𝑥-𝑦 plane

𝑦

𝑧

𝑥

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question

Find the eigenvectors of the following operators:

The operator 𝑂 that projects vectors in ℝ3 onto the 𝑥-𝑦 plane

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝜆2 = 𝜆3 = 1

𝜆1 = 0

𝑣3

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question

Find the eigenvectors of the following operators:

The operator 𝑂 that projects vectors in ℝ3 onto the 𝑥-𝑦 plane

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝜆2 = 𝜆3 = 1

𝜆1 = 0

𝑣3

(all the vectors in the 𝑥-𝑦 plane)

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question

Find the eigenvectors of the following operators:

The operator 𝑂 that mirrors vectors in ℝ3 into the origin

𝑦

𝑧

𝑥

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question

Find the eigenvectors of the following operators:

The operator 𝑂 that mirrors vectors in ℝ3 into the origin

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝑣3

𝜆1 = 𝜆2 = 𝜆3 = −1

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

Question

Find the eigenvectors of the following operators:

The operator 𝑂 that mirrors vectors in ℝ3 into the origin

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝑣3

𝜆1 = 𝜆2 = 𝜆3 = −1(all the vectors in ℝ3)

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝑣3

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

Conclusion:

𝜆1 = 𝜆2 = 𝜆3 = −1(all the vectors in ℝ3)

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝑣3

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

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

Conclusion:

𝜆1 = 𝜆2 = 𝜆3 = −1(all the vectors in ℝ3)

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝑣3

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’ …)

Conclusion:

𝜆1 = 𝜆2 = 𝜆3 = −1(all the vectors in ℝ3)

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝑣3

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’ …)

Conclusion:

𝜆1 = 𝜆2 = 𝜆3 = −1(all the vectors in ℝ3)

𝐻Ψ = 𝐸Ψ

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝑣3

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’ …)

Conclusion:

𝜆1 = 𝜆2 = 𝜆3 = −1(all the vectors in ℝ3)

𝐻Ψ = 𝐸Ψ

𝑎+𝜓𝑛 = 𝑛 + 1 𝜓𝑛+1

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

𝑦

𝑧

𝑥

𝑣1

𝑣2

𝑣3

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’ …)

Conclusion:

𝜆1 = 𝜆2 = 𝜆3 = −1(all the vectors in ℝ3)

𝐻Ψ = 𝐸Ψ

𝑎+𝜓𝑛 = 𝑛 + 1 𝜓𝑛+1

not an eigenstate of 𝑎+

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

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’ …)

Conclusion:

How to find eigenvectors:

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

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’ …)

Conclusion:

How to find eigenvectors:

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

det 𝐴 − 𝜆𝐼 = 0𝐴𝑣 = 𝜆𝑣

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

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’ …)

Conclusion:

How to find eigenvectors:

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

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

det 𝐴 − 𝜆𝐼 = 0𝐴𝑣 = 𝜆𝑣

𝐻Ψ = 𝐸Ψ

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

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’ …)

Conclusion:

How to find eigenvectors:

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

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

det 𝐴 − 𝜆𝐼 = 0𝐴𝑣 = 𝜆𝑣

𝐻Ψ = 𝐸Ψ

if the spectrum is non-degenerate then the eigenfunctions are orthogonal

Introduction to Quantum Mechanics I

Lecture 13: Eigenvalues and eigenfunctions

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’ …)

Conclusion:

How to find eigenvectors:

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

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

det 𝐴 − 𝜆𝐼 = 0𝐴𝑣 = 𝜆𝑣

𝐻Ψ = 𝐸Ψ

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

Reading: Sections 3.3

Summarize section 3.3

Homework due Thursday 9 March :

Lecture 13: Eigenvalues and eigenfunctions