I'm trying to understand the problem of the "inverted" oscillator, which has the following Hamiltonian: ˆH=ˆp22m−kˆx22
Suppose that a particle at the initial time is in a state
Ψ(x)=1(πb2)14exp−x2/2b2
We want to find ⟨Ψ∣ˆx(t)∣Ψ⟩ and ⟨Ψ∣ˆx2(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]=iℏ and Leibniz rule:[A,BC]=[A,B]C+B[A,C]):
iℏ˙ˆx=[ˆx,ˆH]=[ˆx,ˆp22m−kˆx22]=[ˆx,ˆp22m]=ˆpm[ˆx,ˆp]=iℏˆpm
˙ˆx=ˆp/m
Similarly, Heisenberg equation for operator ˆp:
iℏ˙ˆp=[ˆp,ˆH]=[ˆp,ˆp22m−kˆx22]=−[ˆp,kˆx22]=−kˆx[ˆp,ˆx]=iℏkˆx
˙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 ⟨Ψ∣ˆx∣Ψ⟩ and ⟨Ψ∣ˆx2∣Ψ⟩:
⟨Ψ∣ˆx(t)∣Ψ⟩=⟨Ψ∣ˆx(0)cosh(ωt)+ˆp(0)mωsinh(ωt)∣Ψ⟩=⟨Ψ∣ˆx(0)∣Ψ⟩cosh(ωt)+⟨Ψ∣ˆp(0)∣Ψ⟩sinh(ωt)mω
Similarly
⟨Ψ∣ˆx2(t)∣Ψ⟩=⟨Ψ∣ˆx2(0)∣Ψ⟩cosh2(ωt)+⟨Ψ∣ˆp2(0)∣Ψ⟩sinh2(ωt)(mω)2+1mω⟨Ψ∣(ˆx(0)ˆp(0)+ˆp(0)ˆx(0))∣Ψ⟩sinh(ωt)cosh(ωt)
Lets introduce the notation:x0=⟨Ψ∣ˆx(0)∣Ψ⟩, p0=⟨Ψ∣ˆp(0)∣Ψ⟩,x20=⟨Ψ∣ˆx2(0)∣Ψ⟩, p20=⟨Ψ∣ˆp2(0)∣Ψ⟩. Then:
⟨Ψ∣ˆx(t)∣Ψ⟩=x0cosh(ωt)+p0mωsinh(ωt)
⟨Ψ∣^x2(t)∣Ψ⟩=x20cosh2(ωt)+p20sinh2(ωt)(mω)2+1mω⟨Ψ∣(ˆx(0)ˆp(0)+ˆp(0)ˆx(0))∣Ψ⟩sinh(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: ⟨Ψ∣ˆx(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=⟨Ψ∣ˆx∣Ψ⟩=∫Ψ∗xΨdx=∫ΨxΨdx=∫Ψ2xdx
Remember that Ψ(x)=1(πb2)14exp−x2/2b2 :
x0=1(πb2)12∫xexp−x2/b2dx
Now we are going to calculate x20 and p20: x20=⟨Ψ∣ˆx2∣Ψ⟩=∫Ψ∗xΨdx=1(πb2)12∫x2exp−x2/b2dx=1(πb2)12√π(b2)322=b22
p20=⟨Ψ∣ˆp2∣Ψ⟩=−ℏ2∫Ψ∗∂2∂x2Ψdx=ℏ22b2
Finally: ⟨Ψ∣^x2(t)∣Ψ⟩=b22cosh2(ωt)+ℏ22b2sinh2(ωt)(mω)2
Using the equation cosh2(x)−sinh2(x)=1 we get:
⟨Ψ∣^x2(t)∣Ψ⟩=12(b2+ℏ2b2(mω)2)cosh2(ωt)−ℏ22b2(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:
- why ⟨Ψ∣(ˆx(0)ˆp(0)+ˆp(0)ˆx(0))∣Ψ⟩=0 for real Ψ(x)?
- why ⟨Ψ∣ˆx(t)∣Ψ⟩=0?
- 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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