Postulates of Quantum Mechanics

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Page 1 Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj Dawar , Michael Nielsen Jacob Biamonte and students Page 2 Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj Dawar , Michael Nielsen Jacob Biamonte and students Linear Operators • V,W: Vector spaces. • A linear operator A from V to W is a linear function A:V?W. An operator on V is an operator from V to itself. • Given bases for V and W, we can represent linear operators as matrices. • An operator A on V is Hermitian iff it is self-adjoint (A=A † ). • Its diagonal elements are real. Short review Page 3 Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj Dawar , Michael Nielsen Jacob Biamonte and students Linear Operators • V,W: Vector spaces. • A linear operator A from V to W is a linear function A:V?W. An operator on V is an operator from V to itself. • Given bases for V and W, we can represent linear operators as matrices. • An operator A on V is Hermitian iff it is self-adjoint (A=A † ). • Its diagonal elements are real. Short review Eigenvalues & Eigenvectors • v is called an eigenvector of linear operator A iff A just multiplies v by a scalar x, i.e. Av=xv – “eigen” (German) = “characteristic”. • x, the eigenvalue corresponding to eigenvector v, is just the scalar that A multiplies v by. • the eigenvalue x is degenerate if it is shared by 2 eigenvectors that are not scalar multiples of each other. – (Two different eigenvectors have the same eigenvalue) • Any Hermitian operator has all real-valued eigenvectors, which are orthogonal (for distinct eigenvalues). Page 4 Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj Dawar , Michael Nielsen Jacob Biamonte and students Linear Operators • V,W: Vector spaces. • A linear operator A from V to W is a linear function A:V?W. An operator on V is an operator from V to itself. • Given bases for V and W, we can represent linear operators as matrices. • An operator A on V is Hermitian iff it is self-adjoint (A=A † ). • Its diagonal elements are real. Short review Eigenvalues & Eigenvectors • v is called an eigenvector of linear operator A iff A just multiplies v by a scalar x, i.e. Av=xv – “eigen” (German) = “characteristic”. • x, the eigenvalue corresponding to eigenvector v, is just the scalar that A multiplies v by. • the eigenvalue x is degenerate if it is shared by 2 eigenvectors that are not scalar multiples of each other. – (Two different eigenvectors have the same eigenvalue) • Any Hermitian operator has all real-valued eigenvectors, which are orthogonal (for distinct eigenvalues). Exam Problems • Find eigenvalues and eigenvectors of operators. • Calculate solutions for quantum arrays. • Prove that rows and columns are orthonormal. • Prove probability preservation • Prove unitarity of matrices. • Postulates of Quantum Mechanics. Examples and interpretations. • Properties of unitary operators Page 5 Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj Dawar , Michael Nielsen Jacob Biamonte and students Linear Operators • V,W: Vector spaces. • A linear operator A from V to W is a linear function A:V?W. An operator on V is an operator from V to itself. • Given bases for V and W, we can represent linear operators as matrices. • An operator A on V is Hermitian iff it is self-adjoint (A=A † ). • Its diagonal elements are real. Short review Eigenvalues & Eigenvectors • v is called an eigenvector of linear operator A iff A just multiplies v by a scalar x, i.e. Av=xv – “eigen” (German) = “characteristic”. • x, the eigenvalue corresponding to eigenvector v, is just the scalar that A multiplies v by. • the eigenvalue x is degenerate if it is shared by 2 eigenvectors that are not scalar multiples of each other. – (Two different eigenvectors have the same eigenvalue) • Any Hermitian operator has all real-valued eigenvectors, which are orthogonal (for distinct eigenvalues). Exam Problems • Find eigenvalues and eigenvectors of operators. • Calculate solutions for quantum arrays. • Prove that rows and columns are orthonormal. • Prove probability preservation • Prove unitarity of matrices. • Postulates of Quantum Mechanics. Examples and interpretations. • Properties of unitary operators Unitary Transformations • A matrix (or linear operator) U is unitary iff its inverse equals its adjoint: U -1 = U † • Some properties of unitary transformations (UT): – Invertible, bijective, one-to-one. – The set of row vectors is orthonormal. – The set of column vectors is orthonormal. – Unitary transformation preserves vector length: |U?| = |? | • Therefore also preserves total probability over all states: – UT corresponds to a change of basis, from one orthonormal basis to another. – Or, a generalized rotation of? in Hilbert space ? ? = ? i i s 2 2 ) ( Who an when invented all this stuff??Read More

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