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Texts: 1.3 - A realistic interpretation of quantum mechanics: Bohmian mechanics The single-particle Schrödinger equation in the position representation is: h^2 ∂ψ/∂t = - (h^2/2m) ∇^2ψ (The generalization to many particles including entanglement and all sorts of non-local effects is straightforward.) Write the wave function in polar form: ψ = (1/√2π) e^(iθ) where ρ(θ) is the probability density that the particle is at θ, then substitute into the Schrödinger equation, and divide by ψ. (a) Show that taking real and imaginary parts leads to two real equations: (1) ∂ρ/∂t + ∇(ρv) = 0 (2) ∂θ/∂t + (h/2m) (∇θ)^2 + V(θ) = 0 The first equation is a continuity equation: the probability flows with velocity v. The second equation is similar to the classical Hamilton-Jacobi equation with the phase θ playing the role of the Hamilton function and there is an additional quantum potential V(θ). Show that: ∇^2√ρ = - (2m/h^2) V(θ) (b) To interpret these equations, Bohm borrowed an idea from the classical Liouville equation: the reason the probability flows with velocity v is that the particle has a definite position θ and it moves along a smooth trajectory with a definite velocity v. Show that the particle's equation of motion is: dv/dt = - (∇V(θ) + ∇V(θ)) / ρ where ∇·v = 0 is the time derivative along the flow, the convective derivative. Hint: use eq.(2). Remarks: Bohm's mechanics is obviously in agreement with experiment because it is derived from the Schrödinger equation. It is an example of adopting the mathematical formalism of QM and appending an interpretation to it. The interpretation assumes that the particle follows a definite trajectory θ. This is in direct contradiction to the standard (i.e., Copenhagen) interpretation and shows that the latter is not necessary. The Bohmian interpretation provides a counterexample to the inevitability of the Copenhagen interpretation and therein lies its historical importance. The main advantage of the Bohmian approach is ontological clarity: the position of the particle and the wave function are both assumed to be ontic; the wave function is a pilot wave; it guides the motion of the particle. Just as in statistical mechanics, probabilities arise because the initial position within the wave packet is unknown. Several objections can be raised against the Bohmian interpretation: For N particles, the wave function is meant to be real but lives in 3N-dimensional configuration space; the wave influences the motion of the particle but the particle does not react back. The status of the Born rule, |ψ|^2 = ρ, is also a matter of concern: the left-hand side is ontic while the right-hand side is epistemic. And generalizations to relativistic fields have not been satisfactory. Perhaps the most serious objection is that it offers no insights into why the Schrödinger equation takes the particular form it does: Why complex numbers? Why is it linear?

          Texts: 1.3 - A realistic interpretation of quantum mechanics: Bohmian mechanics

The single-particle Schrödinger equation in the position representation is:

h^2 ∂ψ/∂t = - (h^2/2m) ∇^2ψ

(The generalization to many particles including entanglement and all sorts of non-local effects is straightforward.) Write the wave function in polar form:

ψ = (1/√2π) e^(iθ)

where ρ(θ) is the probability density that the particle is at θ, then substitute into the Schrödinger equation, and divide by ψ.

(a) Show that taking real and imaginary parts leads to two real equations:

(1) ∂ρ/∂t + ∇(ρv) = 0
(2) ∂θ/∂t + (h/2m) (∇θ)^2 + V(θ) = 0

The first equation is a continuity equation: the probability flows with velocity v. The second equation is similar to the classical Hamilton-Jacobi equation with the phase θ playing the role of the Hamilton function and there is an additional quantum potential V(θ). Show that:

∇^2√ρ = - (2m/h^2) V(θ)

(b) To interpret these equations, Bohm borrowed an idea from the classical Liouville equation: the reason the probability flows with velocity v is that the particle has a definite position θ and it moves along a smooth trajectory with a definite velocity v. Show that the particle's equation of motion is:

dv/dt = - (∇V(θ) + ∇V(θ)) / ρ

where ∇·v = 0 is the time derivative along the flow, the convective derivative. Hint: use eq.(2).

Remarks: Bohm's mechanics is obviously in agreement with experiment because it is derived from the Schrödinger equation. It is an example of adopting the mathematical formalism of QM and appending an interpretation to it. The interpretation assumes that the particle follows a definite trajectory θ. This is in direct contradiction to the standard (i.e., Copenhagen) interpretation and shows that the latter is not necessary. The Bohmian interpretation provides a counterexample to the inevitability of the Copenhagen interpretation and therein lies its historical importance. The main advantage of the Bohmian approach is ontological clarity: the position of the particle and the wave function are both assumed to be ontic; the wave function is a pilot wave; it guides the motion of the particle. Just as in statistical mechanics, probabilities arise because the initial position within the wave packet is unknown. Several objections can be raised against the Bohmian interpretation: For N particles, the wave function is meant to be real but lives in 3N-dimensional configuration space; the wave influences the motion of the particle but the particle does not react back. The status of the Born rule, |ψ|^2 = ρ, is also a matter of concern: the left-hand side is ontic while the right-hand side is epistemic. And generalizations to relativistic fields have not been satisfactory. Perhaps the most serious objection is that it offers no insights into why the Schrödinger equation takes the particular form it does: Why complex numbers? Why is it linear?

texts 13 a realistic interpretation of quantum mechanics bohmian mechanics the single particle schrodinger equation in the position representation is h2 t h22m 2 the generalization to many p 65865

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