- Def Quantum Physics Problem A Spin 1 2 Particle Represented By An Electron Of M And Charge Q E With E 0 Is At Time 1 (158.21 KiB) Viewed 19 times
def = −e with e ≥ 0 is at time t = 0 sec . in the state |ψ (0)i =
|+i def = |0i+|1i √ 2 . The temporal evolution of the state of the
system is governed by the quantum-mechanical time-dependent
Schr¨odinger equation, i}∂t |ψ (t)i = H ( ˆ t)|ψ (t)i, (2) where }
is the reduced Planck constant, ∂t def = ∂ ∂t , and H is the
Hamiltonian of the quantum system given by, ˆ H ( ˆ t) = −µˆ·B~ (t)
. (3) In Eq. (3), µˆ = (ˆµx, ˆµy, ˆµz) denotes the electron
magnetic moment vector operator and B~ (t) = B0zˆ is the
time-independent external magnetic field vector with B0 a positive
real constant. In SI units, the Hamiltonian in Eq. (3) can be
written as H =ˆ −B0µˆz = −B0 − e} 2m σˆz = e}B0 2m σˆz = h0σˆz, (4)
where the constant h0 in Eq. (4) is h0 def = e}B0 2m with µBohr def
= e} 2m being the so-called Bohr magneton and σˆz is the Pauli
phase-flip operator. Considering the Schr¨odinger equation in (2),
express the unitary quantum propagator Uˆ (t) in terms of the
constant (Hermitian) Hamiltonian in Eq. (4) in order to find the
state of the system |ψ (t)i def = Uˆ (t)|ψ (0)i at an arbitrary
time t ≥ 0. Please, express the single-qubit quantum state |ψ (t)i
∈ H1 2 as a superposition of the Hadamard basis ket states {|+i,
|−i}. What is the probability of finding the electron in the state
|−i def = |0i−|1i √ 2 at the instant t∗ = } h0 π 6 ? What are the
SI-units of the Hamiltonian Hˆ and the Bohr magneton µBohr? What
are the SI-units of h0 and [}ω0]MKSA with ω0 def = eB0 m ? Please,
explain. Please, be neat and precise in your work
def • Quantum Physics Problem. A spin-1/2 particle represented by an electron of m and charge q -e with e > 0 is at time t = 0 sec. in the state |ų (0)) = 1+) def |0}+\2). The temporal evolution of the state of the system is governed by the quantum-mechanical time-dependent Schrödinger equation, iħ04 |(t)) = Ĥ(t) | (t)), (2) where ħ is the reduced Planck constant, of def a at! and ÎN is the Hamiltonian of the quantum system given by, H (t) = -1B (t). (3) = In Eq. (3), h = (ft, flyhez) denotes the electron magnetic moment vector operator and B (t) = Bo2 is the time-independent external magnetic field vector with B, a positive real constant. In SI units, the Hamiltonian in Eq. (3) can be written as = -Boll, = -Bo ( eħ - = eħBo -ô, = hoz. 2m 2m def eħB, with #Bohr def 2m 2m where the constant ho in Eq. (4) is ho em being the so-called Bohr magneton and Ô, is the Pauli phase-flip operator. Considering the Schrödinger equation in (2), express the unitary quantum propagator û (t) in terms of the constant (Hermitian) Hamiltonian in Eq. (4) in order to find the state of the system 4' (t)) def û (t) 4 (0)) at an arbitrary time t > 0. Please, express the single-qubit quantum state 14(t)) E H as a superposition of the Hadamard basis ket states {I+), |->}. What is the probability of finding the electron in the state | -) hom? What are the SI-units of the Hamiltonian în and the Bohr magneton MBohr? What are the SI-units of ho and [hwo]MKSA with wo e eBo? Please, explain. Please, be neat and precise in your work. def 10) - 1) at the instant tu = 2 def ml