Question

Schrödinger's field We consider a complex scalar field $\phi$ with mass m coupled to a potential V along with the action $S = \int d^4x \mathcal{L} = \int d^4x(\frac{i}{2} (\phi^* \partial_t \phi - \phi \partial_t \phi^) - \frac{1}{2m} \vec{\nabla} \phi^ \cdot \vec{\nabla} \phi - V(\vec{x}) |\phi|^2)$ The scalar field $\phi$ in this theory is called the Schrödinger field. a) Show that the resulting equation of motion for the Schrödinger field produces the Schrödinger equation. b) Show that the action has U(1) symmetry. Calculate the resulting Noether current and Noether charge, identify them in the context of quantum mechanics.

          Schrödinger's field
We consider a complex scalar field $\phi$ with mass m coupled to a potential V along with the
action
$S = \int d^4x \mathcal{L} = \int d^4x(\frac{i}{2} (\phi^* \partial_t \phi - \phi \partial_t \phi^*) - \frac{1}{2m} \vec{\nabla} \phi^* \cdot \vec{\nabla} \phi - V(\vec{x}) |\phi|^2)$
The scalar field $\phi$ in this theory is called the Schrödinger field.
a) Show that the resulting equation of motion for the Schrödinger field produces the
Schrödinger equation.
b) Show that the action has U(1) symmetry. Calculate the resulting Noether current and
Noether charge, identify them in the context of quantum mechanics.

Schrödinger's field
We consider a complex scalar field ϕ with mass m coupled to a potential V along with the
action
S = ∫ d^4x ℒ = ∫ d^4x((i)/(2) (ϕ^* ϕ - ϕϕ^*) - (1)/(2m)∇⃗ϕ^* ·∇⃗ϕ - V(x⃗) |ϕ|^2)
The scalar field ϕ in this theory is called the Schrödinger field.
a) Show that the resulting equation of motion for the Schrödinger field produces the
Schrödinger equation.
b) Show that the action has U(1) symmetry. Calculate the resulting Noether current and
Noether charge, identify them in the context of quantum mechanics.

Added by Carmen P.

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