Do we have a potential (quantum) in Bohm quantum mechanics that is associated with the phase factor $S$?
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Answered by
0
Hello mate here is your answer.
Starting with the SE,
iℏ∂Ψ∂t=−ℏ22m∇2Ψ+VΨ,iℏ∂Ψ∂t=−ℏ22m∇2Ψ+VΨ,
and the wave function in polar form of amplitude, RR, and phase, S/ℏS/ℏ,
Ψ=ReiS/ℏ,R,S∈R.Ψ=ReiS/ℏ,R,S∈R.
Substituting this into the SE and separating the real and imaginary parts then yields
∂S∂t=ℏ22m∇2RR−(∇S)22m−V(1)(1)∂S∂t=ℏ22m∇2RR−(∇S)22m−V
and
∂R∂t=−12m(R∇2S+2∇R∇S)(2)(2)∂R∂t=−12m(R∇2S+2∇R∇S)
respectively, where (1), in the classical limit ℏ→0ℏ→0, is the Hamilton-Jacobi equation,
∂S∂t=−H,∂S∂t=−H,
and the null term,
Q=−ℏ22m∇2RR,Q=−ℏ22m∇2RR,
is the quantum potential (which can also be expressed using R=|Ψ|R=|Ψ|).
Now, from (2), using ρ=R2ρ=R2 (Born's rule*), we may also obtain the continuity equation,
∂ρ∂t+∇⋅(ρ∇Sm)=0,∂ρ∂t+∇⋅(ρ∇Sm)=0,
where the probability current,
j⃗ =ρ∇Sm,j→=ρ∇Sm,
is now apparent. Velocity is therefore
v⃗ =∇Sm,v→=∇Sm,
and momentum
p⃗ =∇S.
Hope it helps you.
Starting with the SE,
iℏ∂Ψ∂t=−ℏ22m∇2Ψ+VΨ,iℏ∂Ψ∂t=−ℏ22m∇2Ψ+VΨ,
and the wave function in polar form of amplitude, RR, and phase, S/ℏS/ℏ,
Ψ=ReiS/ℏ,R,S∈R.Ψ=ReiS/ℏ,R,S∈R.
Substituting this into the SE and separating the real and imaginary parts then yields
∂S∂t=ℏ22m∇2RR−(∇S)22m−V(1)(1)∂S∂t=ℏ22m∇2RR−(∇S)22m−V
and
∂R∂t=−12m(R∇2S+2∇R∇S)(2)(2)∂R∂t=−12m(R∇2S+2∇R∇S)
respectively, where (1), in the classical limit ℏ→0ℏ→0, is the Hamilton-Jacobi equation,
∂S∂t=−H,∂S∂t=−H,
and the null term,
Q=−ℏ22m∇2RR,Q=−ℏ22m∇2RR,
is the quantum potential (which can also be expressed using R=|Ψ|R=|Ψ|).
Now, from (2), using ρ=R2ρ=R2 (Born's rule*), we may also obtain the continuity equation,
∂ρ∂t+∇⋅(ρ∇Sm)=0,∂ρ∂t+∇⋅(ρ∇Sm)=0,
where the probability current,
j⃗ =ρ∇Sm,j→=ρ∇Sm,
is now apparent. Velocity is therefore
v⃗ =∇Sm,v→=∇Sm,
and momentum
p⃗ =∇S.
Hope it helps you.
Answered by
3
No, there's only the classical (V), and supplementary quantum (Q) potentials found in Bohmian mechanics (BM). Giving the derivation will hopefully illuminate where this comes from (you may also find Bohm's 1952 papers here and here good to read in this regard), although we do come across other familiar formula involving SS in this process:
Starting with the SE,
iℏ∂Ψ∂t=−ℏ22m∇2Ψ+VΨ,
iℏ∂Ψ∂t=−ℏ22m∇2Ψ+VΨ,
and the wave function in polar form of amplitude, RR, and phase, S/ℏS/ℏ,
Ψ=ReiS/ℏ,R,S∈R.
Ψ=ReiS/ℏ,R,S∈R.
Substituting this into the SE and separating the real and imaginary parts then yields
∂S∂t=ℏ22m∇2RR−(∇S)22m−V(1)
(1)∂S∂t=ℏ22m∇2RR−(∇S)22m−V
and
∂R∂t=−12m(R∇2S+2∇R∇S)(2)
(2)∂R∂t=−12m(R∇2S+2∇R∇S)
respectively, where (1), in the classical limit ℏ→0ℏ→0, is the Hamilton-Jacobi equation,
∂S∂t=−H,
∂S∂t=−H,
and the null term,
Q=−ℏ22m∇2RR,
Q=−ℏ22m∇2RR,
is the quantum potential (which can also be expressed using R=|Ψ|R=|Ψ|).
Now, from (2), using ρ=R2ρ=R2 (Born's rule*), we may also obtain the continuity equation,
∂ρ∂t+∇⋅(ρ∇Sm)=0,
∂ρ∂t+∇⋅(ρ∇Sm)=0,
where the probability current,
j⃗ =ρ∇Sm,
j→=ρ∇Sm,
is now apparent. Velocity is therefore
v⃗ =∇Sm,
v→=∇Sm,
and momentum
p⃗ =∇S.
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