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.