Language: English
Pages: 66
Prerequisites......Page 5
Definition of a linear vector space......Page 7
Linear independence and basis vectors......Page 8
The scalar product......Page 9
Continuous basis functions: Fourier Transforms......Page 11
General orthogonality and completeness in function spaces......Page 12
Example from Quantum Mechanics......Page 13
Domain, Codomain and Range......Page 17
Matrix representations of linear operators......Page 18
Adjoint operator and hermitian operators......Page 20
Eigenvalue equations......Page 21
Sturm-Liouville equations......Page 23
How to bring an equation to SL form......Page 24
Second solutions, singularities......Page 25
Eigenvectors and eigenvalues......Page 26
The quantum-mechanical oscillator and Hermite polynomials......Page 27
Legendre polynomials......Page 28
Bessel functions and the spherical drum......Page 32
First example: Electrostatics......Page 33
The eigenstate method......Page 34
The continuity method......Page 37
Quantum mechanical scattering......Page 38
Time-dependent wave equation......Page 40
Solution for the Green function by Fourier transforms......Page 41
Wave equations in (2+1) dimensions......Page 44
Stationary points......Page 45
Functional of first derivative only......Page 47
No explicit dependence on x......Page 48
One endpoint free......Page 52
More than one function: Hamilton's principle......Page 54
More dimensions: field equations......Page 56
Higher derivatives......Page 59
Lagrange's undetermined multipliers......Page 60
Generalisation to functionals......Page 61
The Rayleigh-Ritz method......Page 64
The Sturm-Liouville equation as a variational problem......Page 66
