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# Postulates of Quantum Mechanics Notes | EduRev

## : Postulates of Quantum Mechanics Notes | EduRev

``` 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
-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??
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