Ammonia molecule in an electric field

$Assumptions={ℏω>0,η>0,0<θ<π/2}

“which side” operator

x=({{1, 0}, {0, -1}})

h=((ℏω)/2)({{0, -1}, {-1, 0}})+(η/2)x

h//MatrixForm

{eval,kets}=Eigensystem[h/.η->ℏω Tan[θ]]//Simplify

h.kets[[1]]==eval[[1]]kets[[1]]//Simplify

bras=Inverse[kets\[Transpose]]

Plot[eval/.ℏω->1,{θ,0,π/2}]

bras[[1]].x.kets[[1]]//Simplify

assume molecule on left at t=0

psi0={0,1}

expand wavefunction in eigenstates of H to get the time dependence

psi=Sum[(E^(-I eval[[n]] t/ℏ))kets[[n]]bras[[n]].psi0,{n,1,2}]//Simplify

amplitude to find particle on left

{0,1}.psi

probability to find particle on left

((1/2) (E^(-((I t ℏω Sec[θ])/(2 ℏ)))) (1-Sin[θ]+(E^((I t ℏω Sec[θ])/ℏ)) (1+Sin[θ])))((1/2) (E^((I t ℏω Sec[θ])/(2 ℏ))) (1-Sin[θ]+(E^((-I t ℏω Sec[θ])/ℏ)) (1+Sin[θ])))//Simplify

%//ExpToTrig//Simplify

%/.ℏω->ℏ ω

pleft=%

check we get our usual tunneling result at zero field

Plot[pleft/.{θ->0,ω->2π},{t,0,5}]

large field--very little tunneling

Plot[pleft/.{θ->0.9 π/2,ω->2π},{t,0,5},PlotRange->{0,1}]

intermediate case--η=ℏω

Plot[pleft/.{θ-> π/4,ω->2π},{t,0,5},PlotRange->{0,1}]