Quantum mechanics Transmission coefficient

using wkb approximation, 1 can obtain tunnelling coefficient looks like

t
=



exp
⁡

(
−
2

∫


x

1





x

2




d
x





2
m


ℏ

2





(
v
(
x
)
−
e
)




)





(
1
+


1
4


exp
⁡

(
−
2

∫


x

1





x

2




d
x





2
m


ℏ

2





(
v
(
x
)
−
e
)




)

)


2





 
,


{\displaystyle t={\frac {\displaystyle \exp \left(-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(v(x)-e\right)}}\,\right)}{\displaystyle \left(1+{\frac {1}{4}}\exp \left(-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(v(x)-e\right)}}\,\right)\right)^{2}}}\ ,}

where




x

1


,


x

2




{\displaystyle x_{1},\,x_{2}}

2 classical turning points potential barrier. in classical limit of other physical parameters larger planck s constant, abbreviated



ℏ
→
0


{\displaystyle \hbar \rightarrow 0}

, transmission coefficient goes zero. classical limit have failed in situation of square potential.

if transmission coefficient less 1, can approximated following formula:

t
≈
16


e

u

0





(
1
−


e

u

0




)

exp
⁡

(
−
2
l





2
m


ℏ

2




(

u

0


−
e
)


)



{\displaystyle t\approx 16{\frac {e}{u_{0}}}\left(1-{\frac {e}{u_{0}}}\right)\exp \left(-2l{\sqrt {{\frac {2m}{\hbar ^{2}}}(u_{0}-e)}}\right)}

where



l
=

x

2


−

x

1




{\displaystyle l=x_{2}-x_{1}}

length of barrier potential.