advanced qm susskind - ramo notes
DESCRIPTION
Review notes on advanced quantum mechanicsTRANSCRIPT
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Advanced Quantum Mechanics - Leonard Susskind
Lectures 7-9 Quantum Fields and their Energy - notes
ramo ([email protected])AP (Calculus BC, Statistics, Physics C, Microeconomics) Teacher
November 10, 2014
Lecture 7.
x| = (x)(x)(x) = P (x) probability of finding particle at x
dx (x)(x) = 1dx i (x)j(x) = ij
i
|i i| = I
y|x =i
y|i i|x
(x y) =i
i(y)i (x)
|n1, n2, . . . , ni, . . . number of particles in each ith statea+i = a
i creation operator
ai = ai annihilation operator
(x) =i
ai i(x)
(x) =i
a+i i (x)
|vacuum = |0 = |0, 0, . . .|x =
i
|i i|xi
i (x)a+i |0 = (x) |0
dx (x)(x) =
dxij
aii (x)ajj(x)
=ij
aiajij =i
aiai =i
Ni total number of particles
currently with Mathematics and Science Department, Beijing Number 2 High School, Beijing
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E =i
Nii
=i
aiaii
Hi = ii[P 2
2m+ V (x)
]i(x) = ii(x)
P = i x[
2
2m+ V (x)
]i(x) = ii(x)
|H | =
dx (x)(
2
2m+ V (x)
)(x)
E =
dx (x)(
2
2m+ V (x)
)(x) [operator]
=
dx
i
a+i i (x)
(
2
2m+ V (x)
)j
aj j(x)
=
dx
ij
a+i i (x)
[
2
2m+ V (x)
]j(x)a
j
=
dx
ij
a+i aj i (x)jj(x)
=ij
a+i aj ijj
operator P = i
dx (x)
x(x) Total momentum of all the particles
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Lecture 8.
(p) =12pi
dp (x)eipx
(x) =12pi
dp (x)eipx
(x) =i
ai i(x)
=12pi
dp a(p)eipx
+(x) =12pi
dp a+(p)eipx
a(p) =12pi
dx (x)eipx
a+(p) =12pi
dx +(x)eipx
[ai , a+j ] = ij
[ai , aj ] = 0
[a+i , a+j ] = 0
[+(x),(y)] = (x y)[+(x),+(y)] = 0
[(x),(y)] = 0
[Re(x),Im(y)] = (x y)
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Lecture 9.
b b
D EF
a
cE D
c
E =
dx (x)
(
2
2m
)(x) + V (x)(x)(x)
E =
dx (x)
(
2
2m
)(x) +mc2(x)(x)
|(t+ ) = (1 iH) |(t)= |(t) iH |(t)
(x) =12pi
dp (p)eipx
(x) =12pi
dq (q)eiqx
dx mc2(x)(x) =
dx mc2
12pi
dq (q)eiqx
12pi
dp (p)eipx
=mc2
2pi
(q)(p)ei(pq)x dq dp dx
=mc2
2pi
(q)(p)(p q) dq dp
=mc2
2pi
(p)(p) dp no change in p
dq (p q)F (q) = F (p)
2
2m(x) =
2
2m
12pi
dp (p)eipx
=12pi
dp (p)
p2
2meipx
=p2
2m(x)
E =
dx (x)
p2
2m(x) + V (x)(x)(x) first term also conserves momentum
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