Saturday, 19 September 2020

"Inverted" quantum oscillator


I'm trying to understand the problem of the "inverted" oscillator, which has the following Hamiltonian: ˆH=ˆp22mkˆx22



Suppose that a particle at the initial time is in a state


Ψ(x)=1(πb2)14expx2/2b2


We want to find Ψˆx(t)Ψ and Ψˆx2(t)Ψ and their asymptotic when t+.


Heisenberg equation for operator ˆx (we use that ˆx does not depend explicitly on time, [ˆx,ˆx]=0, [ˆx,ˆp]=i and Leibniz rule:[A,BC]=[A,B]C+B[A,C]):


i˙ˆx=[ˆx,ˆH]=[ˆx,ˆp22mkˆx22]=[ˆx,ˆp22m]=ˆpm[ˆx,ˆp]=iˆpm


˙ˆx=ˆp/m


Similarly, Heisenberg equation for operator ˆp:


i˙ˆp=[ˆp,ˆH]=[ˆp,ˆp22mkˆx22]=[ˆp,kˆx22]=kˆx[ˆp,ˆx]=ikˆx


˙ˆp=kˆx


So, we have following set of equations: ˙ˆx=ˆp/m; ˙ˆp=kˆx.



Using them, we can write: m¨ˆx=kˆx


Solution of this differential equation is ˆx=Acosh(ωt)+Bsinh(ωt), where ω=km.


Initial conditions: A=ˆx(0); B=˙ˆx(0)ω. Remember that ˙ˆx=ˆp/m. Then B=ˆp(0)mω.


So we have ˆx=ˆx(0)cosh(ωt)+ˆp(0)mωsinh(ωt)


Then ˆx2=ˆx2(0)cosh2(0)+ˆp2(0)(mω)2sinh2(0)+1mω(ˆx(0)ˆp(0)+ˆp(0)ˆx(0))sinh(ωt)cosh(ωt)


Now we want to calculate ΨˆxΨ and Ψˆx2Ψ:


Ψˆx(t)Ψ=Ψˆx(0)cosh(ωt)+ˆp(0)mωsinh(ωt)Ψ=Ψˆx(0)Ψcosh(ωt)+Ψˆp(0)Ψsinh(ωt)mω


Similarly


Ψˆx2(t)Ψ=Ψˆx2(0)Ψcosh2(ωt)+Ψˆp2(0)Ψsinh2(ωt)(mω)2+1mωΨ(ˆx(0)ˆp(0)+ˆp(0)ˆx(0))Ψsinh(ωt)cosh(ωt)


Lets introduce the notation:x0=Ψˆx(0)Ψ, p0=Ψˆp(0)Ψ,x20=Ψˆx2(0)Ψ, p20=Ψˆp2(0)Ψ. Then:



Ψˆx(t)Ψ=x0cosh(ωt)+p0mωsinh(ωt)


Ψ^x2(t)Ψ=x20cosh2(ωt)+p20sinh2(ωt)(mω)2+1mωΨ(ˆx(0)ˆp(0)+ˆp(0)ˆx(0))Ψsinh(ωt)cosh(ωt)


Here is the first big trouble: i've found some lecture notes in the internet, where they say that


Ψ(ˆx(0)ˆp(0)+ˆp(0)ˆx(0))Ψ=0 for real Ψ(x). I tried to prove this statement, but i can't. Can you explain why it is equal to zero?


Well, suppose that this statement is true. Then: Ψˆx(t)Ψ=x0cosh(ωt)+p0mωsinh(ωt)


Ψ^x2(t)Ψ=x20cosh2(ωt)+p20sinh2(ωt)(mω)2


Thus, we expressed values through their meanings at the initial time. So, lets find x0, p0, x20, p20. From this point we will work in the coordinate representation.


x0=ΨˆxΨ=ΨxΨdx=ΨxΨdx=Ψ2xdx


Remember that Ψ(x)=1(πb2)14expx2/2b2 :


x0=1(πb2)12xexpx2/b2dx

We integrate in domains from minus infinity to plus infinity. But this integral is equal to zero! So x0=0
Similarly: p0=ΨˆpΨ=iΨxΨdx=2b2i1(πb2)12xexpx2/b2dx
The domains of integration are the same:from minus infinity to plus infinity. But this is also is equal to zero! So, we have : Ψˆx(t)Ψ=0cosh(ωt)+0mωsinh(ωt)=0
I'm confused by this result! Is there a mistake? If not, can you explain me this result?



Now we are going to calculate x20 and p20: x20=Ψˆx2Ψ=ΨxΨdx=1(πb2)12x2expx2/b2dx=1(πb2)12π(b2)322=b22


p20=Ψˆp2Ψ=2Ψ2x2Ψdx=22b2


Finally: Ψ^x2(t)Ψ=b22cosh2(ωt)+22b2sinh2(ωt)(mω)2


Using the equation cosh2(x)sinh2(x)=1 we get:


Ψ^x2(t)Ψ=12(b2+2b2(mω)2)cosh2(ωt)22b2(mω)2


Thus we see that when t+ dispersion of coordinate Ψ^x2(t)Ψ tends to infinity.


To sum up, I have these several difficulties:



  1. why Ψ(ˆx(0)ˆp(0)+ˆp(0)ˆx(0))Ψ=0 for real Ψ(x)?

  2. why Ψˆx(t)Ψ=0?


  3. why when t+ Ψ^x2(t)Ψ tends to infinity? Also i would be happy if you provide some articles(or books) where i can read about it.




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