The Mathematics of Quantum Mechanics

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 gn be
calculated, given the wave functions and the
expansion basis?
3
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
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 lt ?
(q) ? ?.
A scalar product is denoted by lt ? fgt, 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.
Orthonormal
Base
When the base is orthonormal the expansion
coefficients of the function are calculable by
means of a scalar product
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