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Semiconductor Nanostructures: Quantum states and electronic transport.

Pure spin currents generation in magnetic tunnel junctions by means of adiabatic quantum pumping

Find in Worldcat. So entropy is an adiabatic invariant. In a molecular interpretation, S is the logarithm of the phase space volume of all gas states with energy E T and volume V. The volume of the sphere is.

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Since each gas molecule can be anywhere within the volume V , the volume in phase space occupied by the gas states with energy E is. Since the N gas molecules are indistinguishable, the phase space volume is divided by N! Using Stirling's approximation for the gamma function, and ignoring factors that disappear in the logarithm after taking N large,. For a box of radiation, ignoring quantum mechanics, the energy of a classical field in thermal equilibrium is infinite , since equipartition demands that each field mode has an equal energy on average and there are infinitely many modes.

This is physically ridiculous, since it means that all energy leaks into high frequency electromagnetic waves over time. Still, without quantum mechanics, there are some things that can be said about the equilibrium distribution from thermodynamics alone, because there is still a notion of adiabatic invariance that relates boxes of different size. When a box is slowly expanded, the frequency of the light recoiling from the wall can be computed from the Doppler shift. If the wall is not moving, the light recoils at the same frequency.

If the wall is moving slowly, the recoil frequency is only equal in the frame where the wall is stationary. On the other hand, the energy in the light is also decreased when the wall is moving away, because the light is doing work on the wall by radiation pressure.

The rate at which the pressure does work on the wall is found by multiplying by the velocity:. This means that the change in frequency of the light is equal to the work done on the wall by the radiation pressure.

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The light that is reflected is changed both in frequency and in energy by the same amount:. This function cannot be determined from thermodynamic reasoning alone, and Wien guessed at the form that was valid at high frequency.

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He supposed that the average energy in high frequency modes was suppressed by a Boltzmann-like factor. When the expectation value is added over all modes in a cavity, this is Wien's distribution , and it describes the thermodynamic distribution of energy in a classical gas of photons.

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Wien's Law implicitly assumes that light is statistically composed of packets that change energy and frequency in the same way. The entropy of a Wien gas scales as the volume to the power N , where N is the number of packets. This led Einstein to suggest that light is composed of localizable particles with energy proportional to the frequency. Then the entropy of the Wien gas can be given a statistical interpretation as the number of possible positions that the photons can be in. Suppose that a Hamiltonian is slowly time varying, for example, a one-dimensional harmonic oscillator with a changing frequency.

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  • The action J of a classical orbit is the area enclosed by the orbit in phase space. Since J is an integral over a full period, it is only a function of the energy. The integrand is the Poisson bracket of x and p.

    The Poisson bracket of two canonically conjugate quantities like x and p is equal to 1 in any canonical coordinate system. The Hamiltonian is a function of J only, and in the simple case of the harmonic oscillator. When H has no time dependence, J is constant. When H is slowly time varying, the rate of change of J can be computed by re-expressing the integral for J. This means that for slow variations, there is no lowest order change in the area enclosed by the orbit.

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    This is the adiabatic invariance theorem — the action variables are adiabatic invariants. For a harmonic oscillator, the area in phase space of an orbit at energy E is the area of the ellipse of constant energy,. So if a pendulum is slowly drawn in, so that the frequency changes, the energy changes by a proportional amount. After Planck identified that Wien's law can be extended to all frequencies, even very low ones, by interpolating with the classical equipartition law for radiation, physicists wanted to understand the quantum behavior of other systems.

    The Planck radiation law quantized the motion of the field oscillators in units of energy proportional to the frequency:. Einstein, followed by Debye, extended the domain of quantum mechanics by considering the sound modes in a solid as quantized oscillators. At the Solvay conference , the question of quantizing other motions was raised, and Lorentz pointed out a problem, known as Rayleigh—Lorentz pendulum.

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    • If you consider a quantum pendulum whose string is shortened very slowly, the quantum number of the pendulum cannot change because at no point is there a high enough frequency to cause a transition between the states. But the frequency of the pendulum changes when the string is shorter, so the quantum states change energy.

      Einstein responded that for slow pulling, the frequency and energy of the pendulum both change but the ratio stays fixed. This is analogous to Wien's observation that under slow motion of the wall the energy to frequency ratio of reflected waves is constant.