Mecanica Cuantica: Formulario
Jaime Paredes Peralta
17 de mayo de 2015
Definiciones Elementales
( ~
2
2m2 + V (r)
) = i~
t
ba
dx = P (a x b)
dx = 1
P (j) =N(j)
Nf(j) =
n=0
f(j)P (j) 2j =j2 j2
A = |A | =
Adx A = A(x,i~ x
)
A |n = an |n P (an) = | n | |2
Operadores
r = xex + yey + zez p = i~ H = ~2
2m2 + V
L = r p Lz = ~i
L2 = L2x + L
2y + L
2z
T = ~2
2m2 S S2
Ecuacion de Schrodinger
~2
2m
d2
dx2+ V = E
d
dt= iE
~ = eiEt/~
m |n = mn f(x) =n=1
cn |n cn = n | f(x)
(x, 0) =
n=1
cnn (x, t) =
n=1
cnneiEnt/~
n=1
|cn|2 = 1
Pozo infinito
V (x) =
{0 0 x a otw n =
2
asin(npiax)
En =~2k2n2m
=n2pi2~2
2ma2kn =
npi
an = 1, 2, ...
Oscilador armonico
V (x) =1
2m2 n =
(mpi~
)1/4 12nn!
Hn()e2/2
En =
(n+
1
2
) =
m
~x
a =1
2~m(ip+mx) |n = 1
n!(a+)
no
a+ |n =n+ 1 |n+1 a |n =
n |n1
o =(mpi~
)1/4e
m2~ x
2
En =
(n+
1
2
)~
x =
~
2m(a+ + a) p = i
~m
2(a+ a)
Pozo infinito
V (x) =
{ Vo |x| a0 |x| > a n =
Fex x > a
D cos(lx) 0 < x < a(x) x < 0
=
2mE~
l =
2m(E + Vo)
~En + Vo =
n2pi2~2
2m(2a)2
Conmutadores
[A,B] = AB BC [f(x), p] = i~ dfdx
[AB,C] = A[B,C] + [A,C]B [A,BC] = [A,B]C +B[A,C]
[x, p] = i~ [a, a+] = 1 [ri, pj ] = i~ij[ri, rj ] = 0 [pi, pj ]
= 0
Heisenberg
2A2B
(1
2i[A,B]
)2ES3D
i~
t= H
d3r = 1 n(r, t) = n(r)eiEnt/~
( ~
2
2m2 + V (r)
)(r) = E(r) (r) =
cnn(r)e
iEnt/~
Coordenadas esfericas: (r, , ) = R(r)Y (, )Ecuacion angular
1
[sin
d
d
]+ l(l + 1) sin2 = m2 () = APml (cos)
1
d2
d2= m2 () = eim
Y ml (, ) =
2l + 1
4pi
(l |m|)!(l + |m|)!e
imPml (cos)
Pml (x) = (1 x2)|m|/2(d
dx
)|m|Pl(x)
Pl(x) =1
2ll!
(d
dx
)l(x2 1)l
l = 0, 1, ... m = l,l + 1, ..., l 1, l Y ml = (1)m(Y ml )
2pi0
pi0
[Y ml (, )]Y m
l (, ) sin dd = llmm
Ecuacion Radial (d
dr
[r2d
dr
] 2mr
2
~2[V (r) E]
)R = l(l + 1)R
u(r) = rR(r)( ~
2
2m
d2
dr2+
[V +
~2
2m
l(l + 1)
r2
])u = Eu
Veff = V +~2
2m
l(l + 1)
r2
0
|u|2dr = 1
Atomo de Hidrogeno
V (r) = e2
4pio
1
rEn =
[m
2~2
(e2
4pio
)]1
n2= E1
n2n = 1, 2, ...
E1 = 13.6eV a = 4pio~2
me2= 0.529 1010m
nlm =
(2
na
)3(n l 1)!2n[(n+ l)!]3
er/na(
2r
na
)l[L2l+1nl1(2r/na)
]Y ml (, )
n = 1, 2, ... l = 0, 1, ..., n 1 m = l,l + 1, ..., l 1, l
Lpqp(x) = (1)p(d
dx
)pLq(x)
Lq(x) = ex
(d
dx
)q(exxq)
nlmnlmr
2 sin drdd = nnllmm
E = Ei Ef = 13.6eV(
1
n2i 1n2f
)E = h = c/
1
= R
(1
n2f 1n2i
)R =
m
4pic~3
(e2
4pio
)2= 1.097 107m1
Momentum angular
[Lx, Ly] = i~Lz [Ly, Lz] = i~Lx [Lz, Lx] = i~Ly[Lz, x] = i~y
[L,y] = i~x [Lz, z] = 0
[Lz, px] = i~py [L,py] = i~px [Lz, pz] = 0[Lz, r
2] = 0 [Lz, p2] = 0 [L2,L] = 0
L = Lx iLy [Lz, L] = ~L [L2, L] = 0
LLmp = L2 L2z i(i~LZ)
Eigenfunciones
L =~i
(
1
sin
)Lz =
~i
Lx =~i
( sin
cos cot
)Ly =
~i
(cos
sin cot
)L = ~ei
(
i cot
)L+L = ~2
(2
2+ cot
+ cot2
2
2+ i
)L2 = ~2
(1
sin
(sin
)+
1
sin2
2
2
)
H = E L2 = ~2l(l + 1) Lz = ~m
ES: LzYml (, ) = ~mY ml (, ) L2Y ml (, ) = ~2l(l + 1)Y ml (,
)
1
2mr2
[~2
r
(r2
r
)+ L2
] + V = E
Spin
S2 |s m = ~2s(s+ 1) |s m Sz |s m = ~m |s m
S |s m = ~s(s+ 1)m(m 1) |s (m 1)
s = 0,1
2, 1,
3
2, ... m = s,s+ 1, ..., s 1, s
Spin 1/2 s = 1/2
S2 =3
4~2[1 00 1
]
Sx =~2
[0 11 0
]Sy =
~2
[0 ii 0
]Sz =
~2
[1 00 1
]Spinores (normalizados) de Sx
(x)+ =
[1/
2
1/
2
],
(eval +
~2
)(x) =
[1/
2
1/2],
(eval ~
2
)
=
(a+ b
2
)(x)+ +
(a b
2
)(x)
Spinores (normalizados) de Sy
(y)+ =
12
[1i
],
(eval +
~2
)(y) =
12
[1i],
(eval ~
2
)
=
(a ib
2
)(x)+ +
(a+ ib
2
)(x)
Spinores (normalizados) de Sz
(z)+ =
[10
],
(eval +
~2
)(z) =
[01
],
(eval ~
2
)
= a(x)+ + b
(x)
