Reading: Chapter 1 in Shankar PHY 712 Quantum Mechanics 12-12:50 PM MWF Olin 103 Plan for Lecture 2: Reading: Chapter 1 in Shankar Properties of finite-dimensional linear vector spaces Eigenvalue properties of special operators Singular value decomposition Evaluation of operators and functions of operators 8/30/2017 PHY 741 Fall 2017 -- Lecture 2

Schedule additional office hours by email: natalie@wfu.edu 8/30/2017 PHY 741 Fall 2017 -- Lecture 2

8/30/2017 PHY 741 Fall 2017 -- Lecture 2

8/30/2017 PHY 741 Fall 2017 -- Lecture 2

The notion of linear vector spaces is very useful for describing quantum mechanics. First consider a finite (n) dimensional vector space expressed in terms of an orthonormal basis. 8/30/2017 PHY 741 Fall 2017 -- Lecture 2

8/30/2017 PHY 741 Fall 2017 -- Lecture 2

Example: 8/30/2017 PHY 741 Fall 2017 -- Lecture 2

Special case of “commuting” Hermitian operators 8/30/2017 PHY 741 Fall 2017 -- Lecture 2

Special case of “commuting” Hermitian operators -- continued 8/30/2017 PHY 741 Fall 2017 -- Lecture 2

Special case of “commuting” Hermitian operators -- example 8/30/2017 PHY 741 Fall 2017 -- Lecture 2

Digression – singular value decomposition Some useful properties of matrices 12/02/2016 PHY 711 Fall 2016 -- Lecture 37

Singular value decomposition 12/02/2016 PHY 711 Fall 2016 -- Lecture 37

Singular value decomposition -- continued 12/02/2016 PHY 711 Fall 2016 -- Lecture 37

12/02/2016 PHY 711 Fall 2016 -- Lecture 37

12/02/2016 PHY 711 Fall 2016 -- Lecture 37

Example: = = 12/02/2016 PHY 711 Fall 2016 -- Lecture 37

Note that SVD can be applied to rectangular matrices UT 12/02/2016 PHY 711 Fall 2016 -- Lecture 37

8/30/2017 PHY 741 Fall 2017 -- Lecture 2

Functions of Operators 8/30/2017 PHY 741 Fall 2017 -- Lecture 2

8/30/2017 PHY 741 Fall 2017 -- Lecture 2

Functions of multiple operators 8/30/2017 PHY 741 Fall 2017 -- Lecture 2