Question

A Hamiltonian describing a charged particle (charge ( =e ), mass ( =m ) ) in a constant magnetic field ( B ) in the ( z )-direction is given by [ Hleft(x, y, z, p_{x}, p_{y}, p_{z} ight)=frac{1}{2 m}left(left(p_{x}-frac{e B}{2} y ight)^{2}+left(p_{y}+frac{e B}{2} x ight)^{2}+p_{z}^{2} ight) ] a) Write down the resulting Schrödinger equation for this system (in the position-space representation). b) Use it to obtain the probability current ( vec{J} ) appearing in the conservation law [ frac{partial}{partial t} ho+ abla cdot vec{J}=0, quad ho=|psi|^{2} ]

          A Hamiltonian describing a charged particle (charge ( =e ), mass ( =m ) ) in a constant magnetic field ( B ) in the ( z )-direction is given by
[
Hleft(x, y, z, p_{x}, p_{y}, p_{z}
ight)=frac{1}{2 m}left(left(p_{x}-frac{e B}{2} y
ight)^{2}+left(p_{y}+frac{e B}{2} x
ight)^{2}+p_{z}^{2}
ight)
]
a) Write down the resulting Schrödinger equation for this system (in the position-space representation).
b) Use it to obtain the probability current ( vec{J} ) appearing in the conservation law
[
frac{partial}{partial t} 
ho+
abla cdot vec{J}=0, quad 
ho=|psi|^{2}
]

$A Hamiltonian describing a charged particle (charge ( =e ), mass ( =m ) ) in a constant magnetic field ( B ) in the ( z )-direction is given by [ Hleft(x, y, z, px, py, pz ight)=frac12 mleft(left(px-frace B2 y ight)^2+left(py+frace B2 x ight)^2+pz^2 ight) ] a) Write down the resulting Schrödinger equation for this system (in the position-space representation). b) Use it to obtain the probability current ( vecJ ) appearing in the conservation law [ fracpartialpartial t ho+ abla cdot vecJ=0, quad ho=|psi|^2 ]$

Added by Roy S.

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