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