QM CH3 formalism CH4

QM based on p93
1. wavefuction (represents state) .... vectors
2.operators (represents observables).... linear transformations
linear algebra

in N dimension

a vector
3.1

inner product 3.2


linear transformations

T (specified basis)

3.3

vector (functions)

the integral must converge

3.4

wave functions live in hilbert space

inner product of two functions
3.6
if f, g are square integrable in hilbert space
保證收斂 by
3.7


3.8

3.9


normalize inner product 1

orthogonal inner product 0

orthonormal inner product 3.10

a function is complete if any functions( in hilbert space) can be expressed as linear combination

3.11

if functions are orthonormal coifficients (by fourier tricks)

3.12

p96
{3.2}
observables
hermitian operators

expectation value of observables are next to inner product notation

3.13

real
3.14
conjugate(in inner product) reverse order
3.15

thus for wave functions

3.16 called such operators hermitian ( hermitian 可作用在前或著後)

3.17 by pr 3.3

observables are represents by hermitian operators operators

check
3.19

deteminate state
每次量Q都會不同 可否找到依系統量Q得到 q for const (hamiltonian of stationar state)

in determinate state use deviation=0


3.21
3.22 eigenequation

補依
q is the eigen value
determinate states are eigenfunctions of Q(^)

0 not for eigen function but eigen value

all eigenvalues contituets spectrum

more than 2 linear independent eigenfunctions share same eigenvalue called degenerate

ex 3.1

{3.3}
eifenfuctions of a hermitian operator

eifenfuctions of a hermitian operator (phy determinate state of observables)

1. discrete (eigenvalues are separated)
eifenfuctions lie in hilbert space not reliable wave functions
2. continuous (eigenvalue fill out entire range)
eifenfuctions not normalizable not possible wave functions
可獨立 (hamiltonian oscillator, free particle hamiltonian)
可疊加(hamiltonian for finite square well)
p101
3.3.1 discrete spectra(normalizable eigenfunctions)
1. eigenvalue real


proof
2. eigenfunctions belonging to distinct eigenvalues are orthogonal

proof

axion : the eigenfunctions of an observable operator are complete
any function can be expressed as a linear combination of them

ex3.2
3.31
3.32
3.33
3.34
3.35



CH4

quantum mechanichx in three dimensions

sh eq in spherical coordinates

4.1

H is
4.1...

4.2

4.3

4.4

4.5 laplacian

probability of finding particle in the region

4.6

if V indep of t

4.7

and wavefunction satisfies

the general solution is

4.9

4.1.1

separation of variables

V onlyy dep on x
use spherical coordinates

4.13

sh eq

4.14

separate solution

4.15 to 4.14

divide by RY and ...

xxxxxx

first bracket dep r
others dep thita fi

now add separation constant

4.16
4.17

4.1.2 the angular eq

from 4.17 get 4.18

separate agian
4.19 ???

xxxxxxx

the first term is dep thita second dep fi

4.20
4.21

solution for fi

- absorb by front

4.22

4.23 4.24

for thita

4.25

solution
4.26

associated legendre function
4.27

legendre polynomial from Rodrigues formula

4.28

TABLE

ex

m is odd

l must be nonnegative integer

????
4.29

two solutins (4.25) but the other one blows up for thita=0 or pi

4.23

.......

normalize 4.31

spherical harmonics

4.32
.......
4.33

l is called azimuthal quantum number

m is magnetic quantum number

4.1.3 the radial eq

V(r) affect R(r)
4.16
4.35
4.36
4.37 radial eqation