Postulates of Quantum Mechanics

 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??