1 The Mathematics of Quantum Mechanics 2. Unitary and Hermitian Operators

2 Measurement in Quantum Mechanics Measuring is equivalent to decomposing down the system state to its basis states. What are the basis states for a specific measurement? What values are obtained in the measurement?

3 Rotation Operator The Rotation Group The mathematical generator of the group Group Properties If the change in the function is known as a result of an infinitesimal rotation, the integrated result of applying any rotation operator can be calculated by this infinitesmal change

4 The generator of the rotation group is used to construct a physically meaningful operator The Angular Momentum Operator The result of operating an angular momentum operator on a function is equivalent to a derivation by The result of operating an angular momentum operator on a function is equivalent to a derivation by 

5 The Eigenfunctions and Eigenvalues The eigenvalue equation The eigenfunctions and eigenvalues are obtained as a result of solving the differential equation  The eigenfunctions of the angular momentum operator are the basis states of the measuring of the angular momentum  The eigenvalues of the angular momentum operator are the results obtained by measurement

6 The Link Between the Group Operators and Generator Operators 1.The eigenfunctions are identical 2.The link between the eigenvalues The rotation operator represents a geometrical action that preserves the normalization, and therefore (Unitary Operator) The rotation operator represents a geometrical action that preserves the normalization, and therefore 1 = | | 2 (Unitary Operator) The angular momentum operator represents a measurable physical quantity, and therefore all the eigenvalues are real (Hermitian Operator)

7 The propagation in Time The second postulate of a free particle The second postulate of a two-dimensional rigid rotor Which operator generates the change in time? The Schrödinger equation describes how the system state changes in time:

8 The Evolution Group The Schrödinger time-dependent equation: The Group properties The Group Generator: The Evolution Operator

9 The Hamiltonian - the Energy Operator 1.The dispersion ratio in the second postulate determines the eigenvalue of the evolution operator: The generator of the evolution operator represents the measurement of energy ! 2.An evolution eigenfunction is also a Hamiltonian eigenfunction (and vice versa). Also, the stationary Shrödinger equation should be fulfilled (according to the link between the group operators and the generator):

10 An Example: a Free Particle on a Ring Inserting the appropriate Hamiltonian: Equation of the eigenvalue: Eigenfunctions: Eigenvalues: From classic mechanics From classic mechanics E=L z 2 /2I But also: Obviously is fulfilled

11 1.Finding the basis state - the eigenfunction with the lowest eigenvalue is the most energetically stable state of a chemical system The Significance of the Hamiltonian in Chemistry 2.Sperctroscopy - measuring of energy states. The basis states of the measurment are the eigenfunctions of the Hamiltonian, and the measured values are the appropriate eigenvalues. 3.Dynamical calculations - the eigenfunctions of the Hamiltonian are also eigenfunctions of the evolution operator