1 The Mathematics of Quantum Mechanics 3. State vector, Orthogonality, and Scalar Product.

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Presentation transcript:

1 The Mathematics of Quantum Mechanics 3. State vector, Orthogonality, and Scalar Product

2 Measurement in Quantum Mechanics Measuring is equivalent to breaking the system state down to its basis states: The basis states are eigenfunctions of a hermitian operator: The values that can be obtained by measuring are the eigenvalues, with the following probability: How can the expansion coefficients g n be calculated, given the wave functions and the expansion basis?

3 y x v1v1 v2v2  A Scalar Product in Vectors A scalar product is an action applied to each pair of vectors: Geometrically this means that the length of one vector is multiplied by its projection on of the other one.

4 y x u1u1 u2u2 v An Orthonormal Base in Vectors If Then And in general:

5 State Vectors and Scalar Product (Dirac Notation) Each function is denoted by the state vector: <  (q) |  . A scalar product is denoted by, and fulfills the following conditions:

6 Scalar Product of Functions The scalar product of functions is calculated by: For example, for a particle on a ring:

7 Orthonormal Base Theorem: a set of all the eigenfunctions of a hermitian operator constitutes an orthonormal base. When the base is orthonormal the expansion coefficients of the function are calculable by means of a scalar product: OrthonormalBase

8 Orthonormal Base - a Particle on a Ring Base functions: Orthonormality: Expansion Coefficients (Fourier Theorem)

9 Orthonormal Base - Legendre Plynomials The definition space of the functions is on x axis, in the [1,1-]:  (x)  |  The base functions: Orthonormality: Expansion Coefficients: