A model quantum spin system - Quantum Mechanics

Rtρ¼e^�βH^t⁼²ρe^βH^t⁼²: (84) The paired Hermitian map superoperators act at the start and end of a time interval. They give a measure of the change in the energy of the system over that interval. This procedure does not disturb the system beyond that already incurred by coupling the system to the environment. The Jarzynski inequality now follows by applying this Hermitian map and quantum formalism. Discretize the experi- mental time into a series of discrete intervals labeled by an integer t.

The system Hamiltonian is fixed within each interval. It changes only in discrete jumps at the boundaries. The heat flow can be measured by wrapping the

superoperator time evolution of each time interval Stalong with the corresponding Hermitian map measurementsR^�1_t SRt. In a similar fashion, the measurement of the Boltzmann weighted energy change of the system can be measured with e� ^�^β^ΔE�

¼ TrRτSR^�1_τ . The average Boltzmann weighted work of a driven, dissipative quantum system can be expressed as

e^�^βW

� �

¼ Tr Rτ

t �R^�1_t StRt� R^�1τ ρ^eq₀

, (85)

In (85),ρ^t_eqis the system equilibrium density matrix when the system Hamilto- nian is H^S_t.

This product actually telescopes due to the structure of the energy change Hermitian map (84) and the equilibrium density matrix (65). This leaves only the free energy difference between the initial and final equilibrium ensembles, as can be seen by writing out the first few terms

e^�^βW

� �

¼Tr R^τ�R^�1τ SτR^τ�

⋯�R^�1₂ S₂R2�

R^�1₁ S₁R1

� �

R^�1₀ ρ⁰_eq

h i

¼Tr τ�R^�1τ SτR^τ�

⋯�R^�1₂ S₂R2�

R^�1₁ S₁R1

� � I

Z 0ð Þ

� �

¼Tr½Rτ�R^�1τ SτRτ�

⋯�R^�1₂ S₂R2�

R^�1₁ S₁ρ¹_eq Z 1ð Þ Z 0ð Þ

� �

⋯¼Zð Þτ

Z 0ð Þ¼e^�^β^ΔF¼e^�^β^ΔF:

(86)

In the limit in which the time intervals are reduced to zero, the inequality can be expressed in the continuous Lindblad form:

e^�^β^W

� �

¼TrRð Þt exp ð_t

0Rð Þξ ^�1Sð ÞRξ ð Þξ dξ

� �

Rð Þ0^�1ρ^eq₀ ¼e^�^β^ΔF: (87)

The system Hamiltonian is fixed within each interval. It changes only in discrete jumps at the boundaries. The heat flow can be measured by wrapping the

¼ TrRτSR^�1_τ . The average Boltzmann weighted work of a driven, dissipative quantum system can be expressed as

e^�^βW

� �

¼Tr Rτ

t �R^�1_t StRt� R^�1τ ρ^eq₀

, (85)

In (85),ρ^t_eqis the system equilibrium density matrix when the system Hamilto- nian is H^S_t.

e^�^βW

� �

¼Tr R^τ�R^�1τ SτR^τ�

⋯�R^�1₂ S₂R2�

R^�1₁ S₁R1

� �

R^�1₀ ρ⁰_eq

h i

¼Tr τ�R^�1τ SτR^τ�

⋯�R^�1₂ S₂R2�

R^�1₁ S₁R1

� � I

Z 0ð Þ

� �

¼Tr½Rτ�R^�1τ SτRτ�

⋯�R^�1₂ S₂R2�

R^�1₁ S₁ρ¹_eq Z 1ð Þ Z 0ð Þ

� �

⋯¼Zð Þτ

Z 0ð Þ¼e^�^β^ΔF¼e^�^β^ΔF:

(86)

In the limit in which the time intervals are reduced to zero, the inequality can be expressed in the continuous Lindblad form:

e^�^β^W

� �

¼TrRð Þt exp ð_t

0Rð Þξ^�1Sð ÞRξ ð Þξ dξ

� �

Rð Þ0 ^�1ρ^eq₀ ¼e^�^β^ΔF: (87)

5. A model quantum spin system

A magnetic resonance experiment can be used to illustrate how these ideas can be applied in practice. A sample of noninteracting spin-1=2 particles are placed in a strong magnetic field B0which is directed along the z direction. Denote byσ_j, j¼ x, y, z the usual Pauli matrices and 1 the 2�2 identity matrix. It is assumed the motion of the system is unitary. Then the spin is governed by the Hamiltonian:

H₀¼ �1

2B₀σ_z: (88)

Quantum Mechanics

In units whereℏis one, B0represents the characteristic precession frequency of the spin. Since H_ois diagonal in the∣�ibasis that diagonalizesσ_z, the matrix exponential and partition function are given by

e^�H⁼^T¼ e^B⁰^=2T 0 0 e^�B⁰⁼^2T

, Z¼Tr e�^�H⁼^T�

¼2 cosh B0

� �

, (89)

If we set~σto be the equilibrium magnetization of the system,~σ¼h iσ_x _th, the thermal density matrix is

ρ¼ρ_th¼1 2

1þ~σ 0 0 1�~σ

� �

, ~σ¼ tanh B₀ T

� �

: (90)

and~σcorresponds to the parametric response of a spin-1=2 particle.

The work segment is implemented by introducing a very small field of amplitude B rotating in the xy plane with frequencyω. The work parameter is governed by the field

B¼B sinð ð Þ, cosωt ð Þ, 0ωt Þ: (91) Typically, B0≈ωT and B≈0:01T, so we may approximate B< <B0. The total Hamiltonian is the combination

H tð Þ ¼ �B₀ 2 σ_z�B

2 �σ_zsinð Þ þωt σ_ycosð Þωt �

: (92)

The oscillating field plays the role of a perturbation which although weak may initiate transitions between the up and down spin states and will be most frequent at the resonance conditionω¼B₀, so the driving frequency matches the natural oscillation frequency.

The time evolution operator U tð Þis calculated now. To do this, define a new operator V tð Þby means of the equation

U tð Þ ¼eⁱ^ω^t^σ^z⁼²V tð Þ: (93) Substituting (43) into the evolution equation for U tð Þ, i∂_tU¼H tð ÞU, U 0ð Þ ¼1.

It is found that V tð Þobeys the Schrödinger equation:

i∂V

∂t ¼H t~ð ÞV, V 0ð Þ ¼1, (94) It is found that V tð Þsatisfies

i∂V

∂t ¼1

2 ωσ_z�B₀σ_z�Be^�iωtσ^z⁼²�σ_xsinð Þ þωt σ_ycosð Þωt� e^iωσ^z⁼²

� �

V tð Þ: (95) Using the commutation relations of the Pauli matrices and the fact that

e^�iωσ^z ¼1 cos ωt 2

� �

�iσ_zsin ωt 2

� �, (96)

it is found that the terms in the evolution equation can be simplified e^�i^ασ^zσ_xeⁱ^ασ^z ¼ð1 cosα�iσxsinαÞσ_xð1 cosαþiσzsinαÞ

¼σ_xþ2 sinαcosασ_y�2iσ_zσ_ysin²α¼σ_xþ2 sinαcosασ_ysin²α, (97) Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831

e^�^iασ^zσ_ye^iασ^z¼�σ_ycosα�iσxσ_ysinαð1 cosαþiσzsinαÞ ¼σ_y�2 sinαcosασ_xþ2iσzσsin²α:

(98) By means of these results, it remains to simplify

e^�iωtσ^z⁼²�σ_zsinð Þ þωt σ_ycosð Þωt� e^iωtσ^z⁼²

¼σ_zðsinωt�sinωtþcosωt sinωt�cosωt sinωtÞ þσ_y�sin²ωtþcosωt�cosωtþcos²ωt�

¼σ_y: (98a) Taking these results to (95), we arrive at

i∂V

∂t ¼H₁V, H₁¼ �1

2ðB₀�ωÞσ_z�1

2Bσ_y: (99)

This means V tð Þevolves according to a time-dependent Hamiltonian, so the solution can be written as

V tð Þ ¼e^�iH¹^t, (100) and the full-time evolution operator is given by

U tð Þ ¼eⁱ^ω^t^σ^z⁼²e^�iH¹^t: (101) Since the operatorsσ_yandσ_zdo not commute, the exponentials in (101) cannot be using the usual addition rule.

To express (100) otherwise, suppose M is an arbitrary matrix such that M²¼1.

Whenαis an arbitrary parameter, power series expansion of e^�iαMyields

e^�iαM¼1 cosð Þ �α iM sinð Þ:α (102) Now H1can be put in equivalent form

H₁¼Ω

2�σ_zcosϑþσ_ysinϑ� , Ω¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B0�ω ð Þ²þB² q

, tanϑ¼ B

B₀�ω, (103)

Sinceσ²_i ¼1, it follows that

σ_zcosϑþσ_ysinϑ

� �₂

¼1: (104)

Consequently, (100) can be used to prove that V tð Þis given by e^�iH¹^t¼1 cos Ω

� �

þi�σ_zcosϑþσ_ysinϑ� sin Ω

� �

cos Ω 2t

� �

þi cosϑsin Ω 2t

� �

sinϑsin Ω 2t

� �

�sinϑsin Ω 2t

� �

cos Ω 2t

� �

�i cosϑsin Ω 2t

� �

0 BB B@

1 CC CA

(105)

Since

Quantum Mechanics

e^iωσ^z^t=2¼ eⁱ^ω^t⁼² 0 0 e^�iωt=2

(106)

the evolution operator is then given by

U tð Þ ¼ u tð Þ v tð Þ

�v^∗ð Þt u^∗ð Þt

� �

: (107)

The functions u tð Þand v tð Þin (107) are given as u tð Þ ¼eⁱ^ω^t⁼² cos Ω

� �

þi sinϑsin Ω 2t

� �

, v tð Þ ¼eⁱ^ω^t⁼²�sinϑ� sin Ω 2t

� �

: (108) Apart from a phase factor, the final result depends only onΩandϑ, and these in turn depend on B₀, B, andωthrough (108). To understand the physics of U tð Þa bit better, suppose the system is initially in the pure state∣þi. The probability will be found in state∣�iafter time t is

∣h�∣U tð Þ þij j²¼j jv²: (109) This expression represents the transition probability per unit time a transition will occur. Since the unitarity condition U^†U¼1 implies that uj j²þj jv²¼1, we conclude uj j²is the probability when no transition occurs. Note v is proportional to

sinϑ, which gives a physical meaning toϑ. It represents the transition probability and reaches a maximum whenω¼B₀at resonance whereΩ¼B, so u and v simplify to

u tð Þ ¼e^iωt=2cos B 2t

� �

, v tð Þ ¼e^iωt=2sin B 2t

� �

: (110)

Now that U tð Þis known, the evolution of any observable A can be calculated h iA _t¼Tr U� ^†ð Þt AU tð Þρ�

: (111)

If A is replaced byσ_zin (111), we obtain

σ_z

h i_t¼ Tr u^∗ð Þ �v tt ð Þ v^∗ð Þt u tð Þ

σ_z u tð Þ v tð Þ

�v^∗ð Þt u^∗ð Þt

!1 2

1þ~σ 0 0 1�~σ

¼~σ�j ju²�j jv²�

¼σ~�1�2 vj j²� :

(112)

Substituting vj j², this takes the form σ_z

h i_t¼~σ�cos²ϑþ sin²ϑcosð ÞΩt�

¼ tanh B₀ 2T

� �

cos²ϑþ sin²ϑcosð ÞΩt

� �

: (113) Consider the average work. Suppose B< <B0, so the unperturbed Hamiltonian H0can be used instead of the full Hamiltonian H tð Þwhen expectation values of Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831

e^�^iασ^zσ_ye^iασ^z¼�σ_ycosα�iσxσ_ysinαð1 cosαþiσzsinαÞ ¼σ_y�2 sinαcosασ_xþ2iσzσsin²α:

(98) By means of these results, it remains to simplify

e^�iωtσ^z⁼²�σ_zsinð Þ þωt σ_ycosð Þωt� e^iωtσ^z⁼²

¼σ_zðsinωt�sinωtþcosωt sinωt�cosωt sinωtÞ þσ_y�sin²ωtþcosωt�cosωtþcos²ωt�

¼σ_y: (98a) Taking these results to (95), we arrive at

i∂V

∂t ¼H₁V, H₁¼ �1

2ðB₀�ωÞσ_z�1

2Bσ_y: (99)

This means V tð Þevolves according to a time-dependent Hamiltonian, so the solution can be written as

V tð Þ ¼e^�iH¹^t, (100) and the full-time evolution operator is given by

U tð Þ ¼eⁱ^ω^t^σ^z⁼²e^�iH¹^t: (101) Since the operatorsσ_yandσ_zdo not commute, the exponentials in (101) cannot be using the usual addition rule.

To express (100) otherwise, suppose M is an arbitrary matrix such that M²¼1.

Whenαis an arbitrary parameter, power series expansion of e^�iαMyields

e^�iαM¼1 cosð Þ �α iM sinð Þ:α (102) Now H1can be put in equivalent form

H₁¼Ω

2�σ_zcosϑþσ_ysinϑ� , Ω¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B0�ω ð Þ²þB² q

, tanϑ¼ B

B₀�ω, (103)

Sinceσ²_i ¼1, it follows that

σ_zcosϑþσ_ysinϑ

� �₂

¼1: (104)

Consequently, (100) can be used to prove that V tð Þis given by e^�iH¹^t¼1 cos Ω

� �

þi�σ_zcosϑþσ_ysinϑ� sin Ω

� �

cos Ω 2t

� �

þi cosϑsin Ω 2t

� �

sinϑsin Ω 2t

� �

�sinϑsin Ω 2t

� �

cos Ω 2t

� �

�i cosϑsin Ω 2t

� �

0 BB B@

1 CC CA

(105)

Since

Quantum Mechanics

e^iωσ^z^t=2¼ eⁱ^ω^t⁼² 0 0 e^�iωt=2

(106)

the evolution operator is then given by

U tð Þ ¼ u tð Þ v tð Þ

�v^∗ð Þt u^∗ð Þt

� �

: (107)

The functions u tð Þand v tð Þin (107) are given as u tð Þ ¼eⁱ^ω^t⁼² cos Ω

� �

þi sinϑsin Ω 2t

� �

, v tð Þ ¼eⁱ^ω^t⁼²� sinϑ� sin Ω 2t

� �

sinϑ, which gives a physical meaning toϑ. It represents the transition probability and reaches a maximum whenω¼B₀at resonance whereΩ¼B, so u and v simplify to

u tð Þ ¼e^iωt=2cos B 2t

� �

, v tð Þ ¼e^iωt=2sin B 2t

� �

: (110)

Now that U tð Þis known, the evolution of any observable A can be calculated h iA_t¼ Tr U� ^†ð ÞtAU tð Þρ�

: (111)

If A is replaced byσ_zin (111), we obtain

σ_z

h i_t¼Tr u^∗ð Þ �v tt ð Þ v^∗ð Þt u tð Þ

σ_z u tð Þ v tð Þ

�v^∗ð Þt u^∗ð Þt

!1 2

1þ~σ 0 0 1�~σ

¼σ~�j ju²�j jv²�

¼~σ�1�2 vj j²� :

(112)

Substituting vj j², this takes the form σ_z

h i_t¼σ~�cos²ϑþsin²ϑcosð ÞΩt �

¼ tanh B₀ 2T

� �

cos²ϑþsin²ϑcosð ÞΩt

� �

quantities are calculated which are related to the energy. Let us determine the energy of the system at any t by taking operator A to be H₀:

h i_t ¼ �B0

2 h iσ_z _t¼ �1

2B0Tr U^†ð Þtσ_zU tð ÞρÞ

¼ �1

2B₀Tr u^∗ �v v^∗ u

! 1 0

0 �1

! u v

�v^∗ u^∗

!1 2

1þ~σ 0 0 1�~σ

¼ �B0

4 Tr u^∗ �v

v^∗ u

! u v

v^∗ �u^∗

! 1þ~σ 0 0 1�~σ

¼ �1

4B0~σ�1�2 vj j²� : (114) The average work at time t is simply the difference between the energy st time t1

and t¼0. Since v 0ð Þ ¼0, this difference is h iW _t ¼ �B₀

2 ~σ�1�2 vj j²� þB₀

2 ~σ¼~σB0j jv²

¼~σB₀sin²ϑsin² Ω 2t

� �

¼~σB₀ B

Ω² sin² Ω 2t

� �

(115)

since sin²ϑ¼1�cos²ϑ¼B²=Ω². The average work oscillates indefinitely with frequencyΩ=2. This is a consequence of the fact the time evolution is unitary. The amplitude multiplying the average work is proportional to the initial magnetization

~σand B²=Ω², so the ratio is a Lorentzian function.

The equilibrium free energy is F¼ �T log Z where Z¼2 cosh Bð 0=2TÞ. The free energy of the initial state at t¼0 and final state at any arbitrary time is the same yielding

ΔF¼0: (116)

This is a consequence of the fact that B< <B0. According to Wh i≥F, it should be expected that

h iW _t≥ ΔF ¼0: (117)

Given the matrices for U tð Þandρthat have been determined so far, the function Gcan be computed:

Gð Þ ¼y Tr U� ^†ð Þey ^ixH^fU yð Þe^�^iyHⁱρ�

¼Tr u^∗ �v v^∗ u

! e^�^iyB⁰⁼² 0 0 e^iyB⁰⁼²

! u v

�v^∗ u^∗

! e^iyB⁰⁼² 0 0 e^�iyB⁰⁼²

! 1

2ð1þ~σÞ 0

0 1

2ð1�σ~Þ 0

1 CC A

¼j ju²þ1

2 ð1þ~σÞe^iyB⁰þ�1�~σe^�iyB⁰� j jv²:

�

(118) Set x¼iβand recall use definition (42) for~σin the second term of (118) to give

1þ tanh β 2B0

� �

e^�^β^B⁰þ 1�tanh β 2B0

� �

e^β^B⁰

¼2 coshðβB0Þ �2 tanh β 2B0

� �

sinhðβB0Þ ¼1: (119) Quantum Mechanics

Substituting (119) into (118), we can conclude e^�^β^W

� �

¼ Gð Þ ¼iβ j ju²þj jv²¼1: (120) This is the Jarzynski inequality, since it is the case thatΔF¼0 here. The statistical moments of the work can be obtained by writing an expression forGinto a power series

Gð Þ ¼y j ju²þj jv² 1þi~σB₀y�1

2B²₀y²þ⋯

� �

: (121)

From (121), the first and second moments can be obtained; for example h i ¼W ~σB0j jv², �W²�

¼B²₀j jv²: (122) As a consequence, the variance of the work can be determined

var Wð Þ ¼�W²�

�h iW ²¼B²₀j jv²�~σ²B²₀j jv⁴�B²₀j jv²�1�~σ²j jv²�

: (123)

A final calculation that may be considered is the full distribution of workPð Þ.W NowPð ÞW is the inverse Fourier transform ofGð Þ:y

Pð Þ ¼W 1 2π

ð_∞

�∞dyGð Þey ^�iyW: (124) Using the Fourier integral form of the delta function, (124) can be written as

Pð Þ ¼W 1 2π

ð_∞

�∞dyGð Þey ^�iyW

¼ 1 2π

ð_∞

�∞dy u tj ð Þj²þjv tð Þj² 1

2ð1þ~σÞe^iB⁰^yþ1

2ð1�~σÞe^�iB⁰^y

� �

e^�iyW:

¼j ju²δð Þ þW 1

2jv tð Þj²δðW�B0Þ þ1

2jv tð Þj²ð1þ~σÞδðWþB0Þ:

(125)

Work taken as a random variable can take three values W ¼0, þB0, �B0

where B0is the energy spacing between the up and down states. The event W ¼B0corresponds to the case where the spin was originally up and then reversed, so an up-down transition. The energy change is Bð 0=2Þ � �ð B₀=2Þ ¼ B₀. Similarly, W¼ �B₀is the opposite flip from this one, and W¼0 is the case with no spin flip.

The second law would have us think that W>0, but a down-up flip should have W ¼ �B₀, soPðW ¼ �B₀Þis the probability of observing a local violation of the second law. However, sincePðW ¼ �B₀Þis proportional to 1�~σ, up-down flips are always more likely than down-up. This ensures that Wh i≥0, so violations of the second law are always exceptions to the rule and never dominate.

The work performed by an external magnetic field on a single spin-1=2 particle has been studied so far. The energy differences mentioned correspond to the work.

For noninteracting particles, energy is additive. Hence the total workh iW which is performed during a certain process is the sum of works performed on each individual particleW ¼W1þ⋯þWN. Since the spins are all independent and energy is an extensive variable, it follows thath i ¼W N Wh i. where Wh iis the average work from (115).

Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831

quantities are calculated which are related to the energy. Let us determine the energy of the system at any t by taking operator A to be H₀:

h i_t¼ �B0

2 h iσ_z _t ¼ �1

2B0Tr U^†ð Þtσ_zU tð ÞρÞ

¼ �1

2B₀Tr u^∗ �v v^∗ u

! 1 0

0 �1

! u v

�v^∗ u^∗

!1 2

1þ~σ 0 0 1�σ~

¼ �B0

4 Tr u^∗ �v v^∗ u

! u v

v^∗ �u^∗

! 1þ~σ 0 0 1�~σ

¼ �1

4B0~σ�1�2 vj j²� : (114) The average work at time t is simply the difference between the energy st time t1

and t¼0. Since v 0ð Þ ¼0, this difference is h iW _t¼ �B₀

2 σ~�1�2 vj j²� þB₀

2 ~σ¼~σB0j jv²

¼~σB₀sin²ϑsin² Ω 2t

� �

¼σB~ ₀ B

Ω² sin² Ω 2t

� �

(115)

The equilibrium free energy is F¼ �T log Z where Z ¼2 cosh Bð 0=2TÞ. The free energy of the initial state at t¼0 and final state at any arbitrary time is the same yielding

ΔF¼0: (116)

This is a consequence of the fact that B< <B0. According to Wh i≥F, it should be expected that

h iW _t≥ ΔF¼0: (117)

Given the matrices for U tð Þandρthat have been determined so far, the function Gcan be computed:

Gð Þ ¼y Tr U� ^†ð Þey ^ixH^fU yð Þe^�^iyHⁱρ�

¼Tr u^∗ �v v^∗ u

! e^�^iyB⁰⁼² 0 0 e^iyB⁰⁼²

! u v

�v^∗ u^∗

! e^iyB⁰⁼² 0 0 e^�iyB⁰⁼²

! 1

2ð1þ~σÞ 0

0 1

2ð1�~σÞ 0

1 CC A

¼j ju²þ1

2 ð1þσ~Þe^iyB⁰þ�1�~σe^�iyB⁰� j jv²:

�

(118) Set x¼iβand recall use definition (42) forσ~in the second term of (118) to give

1þtanh β 2B0

� �

e^�^β^B⁰ þ 1� tanh β 2B0

� �

e^β^B⁰

¼2 coshðβB0Þ �2 tanh β 2B0

� �

sinhðβB0Þ ¼1: (119) Quantum Mechanics

Substituting (119) into (118), we can conclude e^�^β^W

� �

Gð Þ ¼y j ju²þj jv² 1þi~σB₀y�1

2B²₀y²þ⋯

� �

: (121)

From (121), the first and second moments can be obtained; for example h i ¼W σB~ 0j jv², �W²�

¼B²₀j jv²: (122) As a consequence, the variance of the work can be determined

var Wð Þ ¼�W²�

�h iW ²¼B²₀j jv²�~σ²B²₀j jv⁴�B²₀j jv²�1�~σ²j jv²�

: (123)

A final calculation that may be considered is the full distribution of workPð Þ.W NowPð ÞW is the inverse Fourier transform ofGð Þ:y

Pð Þ ¼W 1 2π

ð_∞

�∞dyGð Þey ^�iyW: (124) Using the Fourier integral form of the delta function, (124) can be written as

Pð Þ ¼W 1 2π

ð_∞

�∞dyGð Þey ^�iyW

¼ 1 2π

ð_∞

�∞dy u tj ð Þj²þjv tð Þj² 1

2ð1þ~σÞe^iB⁰^yþ1

2ð1�~σÞe^�iB⁰^y

� �

e^�iyW:

¼j ju²δð Þ þW 1

2jv tð Þj²δðW�B0Þ þ1

2jv tð Þj²ð1þ~σÞδðWþB0Þ:

(125)

Work taken as a random variable can take three values W¼0, þB0, �B0

where B0is the energy spacing between the up and down states. The event W¼B0corresponds to the case where the spin was originally up and then reversed, so an up-down transition. The energy change is Bð 0=2Þ � �ð B₀=2Þ ¼ B₀. Similarly, W¼ �B₀is the opposite flip from this one, and W¼0 is the case with no spin flip.

The second law would have us think that W>0, but a down-up flip should have W¼ �B₀, soPðW¼ �B₀Þis the probability of observing a local violation of the second law. However, sincePðW ¼ �B₀Þis proportional to 1�~σ, up-down flips are always more likely than down-up. This ensures that Wh i≥0, so violations of the second law are always exceptions to the rule and never dominate.

The work performed by an external magnetic field on a single spin-1=2 particle has been studied so far. The energy differences mentioned correspond to the work.

Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831

6. Conclusions

We have tried to give an introduction to this frontier area that lies in between that of thermodynamics and quantum mechanics in such a way as to be compre- hensible. There are many other areas of investigation presently which have had interesting repercussions for this area as well. There is a growing awareness that entanglement facilitates reaching equilibrium [21–23]. It is then worth mentioning that the ideas of einselection and entanglement with the environment can lead to a time-independent equilibrium in an individual quantum system and statistical mechanics can be done without ensembles. However, there is really a lot of work yet to be done in these blossoming areas and will be left for possible future expositions.

Author details Paul Bracken

Department of Mathematics, University of Texas, Edinburg, TX, USA

*Address all correspondence to: [email protected]

by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Quantum Mechanics

References

[1]Boltzmann L. Vorlesungen über Gastheorie. Leipzig: Barth; 1872 [2]Landau L, Lifschitz E. Statistical Mechanics. Oxford Press, Oxford; 1978 [3]Gibbons E, Hawking S. Euclidean Quantum Gravity. Singapore: World Scientific; 1993

[4]Lebowitz J. From time-symmetric microscopic dynamics to time- asymmetric macroscopic behavior an overview. In: Gallavotti G, Reuter W, Yngyason J, editors. Boltzmann’s Legacy. Zürich: European Mathematical Society; 2008

[5]Bracken P. A quantum version of the classical Szilard engine. Central European Journal of Physics. 2014;12:1-8 [6]Bracken P. Quantum dynamics, entropy and quantum versions of Maxwell’s demon. In: Bracken P, editor.

Recent Advances in Quantum

Dynamics. Croatia: IntechOpen; 2016.

pp. 241-263

[7]Fermi E. Thermodynamics. Mineola, NY: Dover; 1956

[8]Allahverdyan A, Balian R, Nieuwenhuisen T. Understanding quantum measurement from the solution of dynamical models. Physics Reports. 2013;525:1-166

[9]Narnhofer H, Wreszinski WF. On reduction of the wave packet, decoherence, irreversibility and the second law of thermodynamics. Physics Reports. 2014;541:249-273

[10]Linblad G. Entropy, information and quantum mechanics.

Communications in Mathematical Physics. 1973;33:305-322

[11]Lindblad G. Expectations and entropy inequalities for finite quantum

systems. Communications in

Mathematical Physics. 1974;39:111-119 [12]Lindblad G. Completely positive maps and entropy inequalities. Communications in Mathematical Physics. 1973;40:147-151

[13]Lieb E. The stability of matter. Reviews of Modern Physics. 1976;48: 553-569

[14]Fano U. Description of states in quantum mechanics and density matrix techniques. Reviews of Modern Physics. 1957;29:74-93

[15]Ribeiro W, Landi GT, Semião F. Quantum thermodynamics and work fluctuations with applications to magnetic resonance. American Journal of Physics. 2016;84:948-957

[16]Wehrl A. General properties of entropy. Reviews of Modern Physics. 1978;50:221-260

[17]Bracken P. A quantum Carnot engine in three dimensions. Advanced Studies in Theoretical Physics. 2014;8: 627-633

[18]Lieb E. The classical limit of quantum spin systems.

Communications in Mathematical Physics. 1973;31:327-340

[19]Lieb E, Liebowitz J. The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei. Advances in Mathematics. 1972; 9:316-398

[20]Crooks J. On the Jarzynski relation for dissipative systems. Journal of Statistical Mechanics: Theory and Experiment. 2008;10023:1-9 [21]Vedral V. The role of relative entropy in quantum information theory. Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics

DOI: http://dx.doi.org/10.5772/intechopen.91831

6. Conclusions

Author details Paul Bracken

Department of Mathematics, University of Texas, Edinburg, TX, USA

*Address all correspondence to: [email protected]

by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Quantum Mechanics

References

Recent Advances in Quantum

Dynamics. Croatia: IntechOpen; 2016.

pp. 241-263

[7]Fermi E. Thermodynamics. Mineola, NY: Dover; 1956

[8]Allahverdyan A, Balian R, Nieuwenhuisen T. Understanding quantum measurement from the solution of dynamical models. Physics Reports. 2013;525:1-166

[9]Narnhofer H, Wreszinski WF. On reduction of the wave packet, decoherence, irreversibility and the second law of thermodynamics. Physics Reports. 2014;541:249-273

[10]Linblad G. Entropy, information and quantum mechanics.

Communications in Mathematical Physics. 1973;33:305-322

[11]Lindblad G. Expectations and entropy inequalities for finite quantum

systems. Communications in

Mathematical Physics. 1974;39:111-119 [12]Lindblad G. Completely positive maps and entropy inequalities.

Communications in Mathematical Physics. 1973;40:147-151

[13]Lieb E. The stability of matter.

Reviews of Modern Physics. 1976;48:

553-569

[14]Fano U. Description of states in quantum mechanics and density matrix techniques. Reviews of Modern Physics.

1957;29:74-93

[15]Ribeiro W, Landi GT, Semião F.

Quantum thermodynamics and work fluctuations with applications to magnetic resonance. American Journal of Physics. 2016;84:948-957

[16]Wehrl A. General properties of entropy. Reviews of Modern Physics.

1978;50:221-260

[17]Bracken P. A quantum Carnot engine in three dimensions. Advanced Studies in Theoretical Physics. 2014;8:

627-633

[18]Lieb E. The classical limit of quantum spin systems.

Communications in Mathematical Physics. 1973;31:327-340

[19]Lieb E, Liebowitz J. The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei. Advances in Mathematics. 1972;

9:316-398

Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831

Dalam dokumen Quantum Mechanics (Halaman 103-119)