The hamiltonian of a spin in a magnetic field is given by a(bbysy+b, s), where α and the three components of b are all constants. (a) compute the energies and corresponding eigenstates of this system (b) let by-0 and b = 0. (i) use the energies and eigenstates for this case to determine the time evolution ψ(t) of a state with initial condition ψ(0) (ii) calculate the expectation values 〈s) and (s) as a function of tine. (iii) use a taylor expansion to show that the time-evolution operator u(t) exp(-iht/h) can be written as (e)s()i - isin(ut)x, where i is the identity operator, x (2/h)s, and w ab/2. hint: use the pauli matrix representation of the spin operator and show that

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explanation:

d. air pressure decreases as altitude increases.

scattering of light by tiny particles in air to produce the colors we see on sky, is dependent on the following factors:

(i) wavelength of light

(ii) size of the scattering particle  

(iii) distance traveled by light through the atmosphere.

according to rayleigh's law of scattering, the amount of scattering is inversely proportional to the fourth power of its wavelength. shorter the wavelength, greater is its scattering. hence violet/blue light is scattered more than red.

sun's light is a composite light ( visible light) with wavelengths in the range 400 nm (violet) to 700 nm (red). as sunlight passes through our atmosphere, the molecules of air scatter light according to their wavelengths. during sunrise and sunset, the sun's position is below the horizon. hence light from the sun travels a greater distance in the atmosphere when compared to its position a noon.

as sunlight enters the earth's atmosphere, blue light is scattered away and at the end of its journey in the sky, only red and orange wavelengths remain in the beam.

hence at sunrise and sunset, the sky appears reddish orange.