Introduction to Quantum Mechanics I Lecture 13...
Transcript of Introduction to Quantum Mechanics I Lecture 13...
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Introduction to Quantum Mechanics I
Lecture 13: Eigenvalues and eigenfunctions
Eigenvalues and eigenfunctions
![Page 2: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/2.jpg)
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
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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
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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
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Introduction to Quantum Mechanics I
OperatorsToday:
What are operators?
Observables?
Hermitian operators?
Determinate states?
What is a degenerate spectrum?
Lecture 13: Eigenvalues and eigenfunctions
![Page 6: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/6.jpg)
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
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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
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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
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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
![Page 10: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/10.jpg)
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
![Page 11: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/11.jpg)
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
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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
![Page 13: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/13.jpg)
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
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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
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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
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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)
![Page 17: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/17.jpg)
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
![Page 18: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/18.jpg)
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
![Page 19: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/19.jpg)
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
![Page 20: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/20.jpg)
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
![Page 21: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/21.jpg)
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
![Page 22: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/22.jpg)
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
![Page 23: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/23.jpg)
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)
![Page 24: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/24.jpg)
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
![Page 25: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/25.jpg)
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
![Page 26: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/26.jpg)
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)
![Page 27: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/27.jpg)
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)
![Page 28: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/28.jpg)
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)
![Page 29: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/29.jpg)
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)
![Page 30: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/30.jpg)
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
![Page 31: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/31.jpg)
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
![Page 32: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/32.jpg)
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+
![Page 33: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/33.jpg)
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:
![Page 34: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/34.jpg)
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
![Page 35: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/35.jpg)
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
![Page 36: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/36.jpg)
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
![Page 37: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/37.jpg)
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
![Page 38: Introduction to Quantum Mechanics I Lecture 13 ...faculty.uml.edu/Johannes_Zwanikken/Documents/Lectures Quantum I... · Lecture 13: Eigenvalues and eigenfunctions ... Part III Quantum](https://reader034.fdocuments.net/reader034/viewer/2022051509/5b6f90877f8b9a58578c5a60/html5/thumbnails/38.jpg)
Introduction to Quantum Mechanics I
Reading: Sections 3.3
Summarize section 3.3
Homework due Thursday 9 March :
Lecture 13: Eigenvalues and eigenfunctions