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a quick summary of WKB method in quantum physics
Typology: Summaries
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turning
points ID
i
↑
i
for
radial (^) parts.
↓
&
bound state^ energy
and un >
classical region
Q =^ Prob(x)^
tunneling
rates
where is^ higher^
↑(2)
= (^) A etikx (^) k= vm(E- (^) v)/t
of
vixs not^ constant^ but^ slowly varying
4(x)
= (^) A(x)e[
P(x)
↑ (x)^ close^ to^ Ike
Similarly
, if
4(x)
= (^) A etkx^ k
= xm(V
but
turning
points (^) require special
treatment.
9 ./ (^) E > V region ,^
classical
V()
= El
=
P(x) = 25E-VA]
4(x)
= (^) A(x)eib(x)
=(A^
classical momentum
= [A"+ (^) ziq+^ [AP"
=
A
Re :^ A"-AIP-AE All
=
= (^0) or (^) (AP)
= (^0)
=> A
= F p
C is^ a (^) const.
Assume
1 and
↑(x)
= []p(x)^
dx
absorb th into
Y(x)
=
Sp()dx
~
Tunneling
for
for
po (^) imaginary ,^
the (^) derivation (^) is the (^) same
4 (2)^
e
I-[ S/p(x) Ide
1
considerA
5 =^ -
--
O (^) A
4 (2)
= (^) A zikx
= (^) Feikx
+Be-
k =^ v -^2 mE /h
transmission prob (^) =
in the^ tunneling region^
4 (2)^
SpIda^
I
-e
E
If
the (^) harrier is^ high
and wide (^).^ 7)
C (^) musthe^ small^
She h
e. g.
2 Alpha (^) decay.
↑ (^) conbomb
I -^ -- & potential
E
Vi ra Y
=
E is^ the emitted &t energy
= -Es
de
= Eme (^) N-1 dr
let u=^1 nu
=
for
VV
]
=
= k^ , - - k2NY NE
, = (E)
= (^1).^ 980MeV
(
= 1.^ 485 fn
2 the^ prob.
ar,
each (^) collision
: The^ mob
of
emission per
unit
-^18
time is^ e
28
Oh
--
lifeline
= (^) e
T
I
> E
in
region
2
(01px)(d =^
(x)
=
=
D
bo
in region
Spa (
with (^) plc =
A
↑()= 24(x) [Bei3(
4p(x)
=
( 2x)xsm(
> (- 2x)
F
=
v (
=> (^) B =^ - jeiπ/D C^ =^ je^
D
if
not at^ x= 0 but at^ x=^22
Eg
D 4 (0)^ = (^0) ·
. p()dx^
E = (^) n or (^) Jopdx =^ (n - = Nh eg , for half H-^ O. V(x) = Simwise xEO
MwN I x= E N ~
> ... Odd mode (x (^) p(x)dx= mw( dx =^ mux= E D 2w only
(^1) , , (^) ... chw Eg
. (^4)
approaches for^ [
for- C by
or try ,
(note : (^) D =D' (^) or D^ =^ - ja (^) (d = (^) In-EI n =^1 , 2
Guoy Phase !!
