Download presentation
Presentation is loading. Please wait.
Published byValentine Lewis Modified over 9 years ago
1
1 The Mathematics of Quantum Mechanics 3. State vector, Orthogonality, and Scalar Product
2
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
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
4 y x u1u1 u2u2 v An Orthonormal Base in Vectors If Then And in general:
5
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
6 Scalar Product of Functions The scalar product of functions is calculated by: For example, for a particle on a ring:
7
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
8 Orthonormal Base - a Particle on a Ring Base functions: Orthonormality: Expansion Coefficients (Fourier Theorem)
9
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:
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.