5.10: Appendix II- Ideal Bose Gas Condensation
- Page ID
- 18767
\( \newcommand\bes{\begin{equation}\begin{split}}\)
\( \newcommand\ltwid{\propto}\)
\( \newcommand\ees{\end{split}\end{equation}}\)
\( \newcommand\mib{\mathbf}\)
\( \newcommand\Sa{\textsf a}\)
\( \newcommand\Sb{\textsf b}\)
\( \newcommand\Sc{\textsf c}\)
\( \newcommand\Sd{\textsf d}\)
\( \newcommand\Se{\textsf e}\)
\( \newcommand\Sf{\textsf f}\)
\( \newcommand\Sg{\textsf g}\)
\( \newcommand\Sh{\textsf h}\)
\( \newcommand\Si{\textsf i}\)
\( \newcommand\Sj{\textsf j}\)
\( \newcommand\Sk{\textsf k}\)
\( \newcommand\Sl{\textsf l}\)
\( \newcommand\Sm{\textsf m}\)
\( \newcommand\Sn{\textsf n}\)
\( \newcommand\So{\textsf o}\)
\( \newcommand\Sp{\textsf p}\)
\( \newcommand\Sq{\textsf q}\)
\( \newcommand\Sr{\textsf r}\)
\( \newcommand\Ss{\textsf s}\)
\( \newcommand\St{\textsf t}\)
\( \newcommand\Su{\textsf u}\)
\( \newcommand\Sv{\textsf v}\)
\( \newcommand\Sw{\textsf w}\)
\( \newcommand\Sx{\textsf x}\)
\( \newcommand\Sy{\textsf y}\)
\( \newcommand\Sz{\textsf z}\)
\( \newcommand\SA{\textsf A}\)
\( \newcommand\SB{\textsf B}\)
\( \newcommand\SC{\textsf C}\)
\( \newcommand\SD{\textsf D}\)
\( \newcommand\SE{\textsf E}\)
\( \newcommand\SF{\textsf F}\)
\( \newcommand\SG{\textsf G}\)
\( \newcommand\SH{\textsf H}\)
\( \newcommand\SI{\textsf I}\)
\( \newcommand\SJ{\textsf J}\)
\( \newcommand\SK{\textsf K}\)
\( \newcommand\SL{\textsf L}\)
\( \newcommand\SM{\textsf M}\)
\( \newcommand\SN{\textsf N}\)
\( \newcommand\SO{\textsf O}\)
\( \newcommand\SP{\textsf P}\)
\( \newcommand\SQ{\textsf Q}\)
\( \newcommand\SR{\textsf R}\)
\( \newcommand\SS{\textsf S}\)
\( \newcommand\ST{\textsf T}\)
\( \newcommand\SU{\textsf U}\)
\( \newcommand\SV{\textsf V}\)
\( \newcommand\SW{\textsf W}\)
\( \newcommand\SX{\textsf X}\)
\( \newcommand\SY{\textsf Y}\)
\( \newcommand\SZ{\textsf Z}\)
\( \newcommand\Ha{\hat a}\)
\( \newcommand\Hb{\hat b}\)
\( \newcommand\Hc{\hat c}\)
\( \newcommand\Hd{\hat d}\)
\( \newcommand\He{\hat e}\)
\( \newcommand\Hf{\hat f}\)
\( \newcommand\Hg{\hat g}\)
\( \newcommand\Hh{\hat h}\)
\( \newcommand\Hi{\hat \imath}\)
\( \newcommand\Hj{\hat \jmath}\)
\( \newcommand\Hk{\hat k}\)
\( \newcommand\Hl{\hat l}\)
\( \newcommand\Hm{\hat m}\)
\( \newcommand\Hn{\hat n}\)
\( \newcommand\Ho{\hat o}\)
\( \newcommand\Hp{\hat p}\)
\( \newcommand\Hq{\hat q}\)
\( \newcommand\Hr{\hat r}\)
\( \newcommand\Hs{\hat s}\)
\( \newcommand\Ht{\hat t}\)
\( \newcommand\Hu{\hat u}\)
\( \newcommand\Hv{\hat v}\)
\( \newcommand\Hw{\hat w}\)
\( \newcommand\Hx{\hat x}\)
\( \newcommand\Hy{\hat y}\)
\( \newcommand\Hz{\hat z}\)
\( \newcommand\HA{\hat A}\)
\( \newcommand\HB{\hat B}\)
\( \newcommand\HC{\hat C}\)
\( \newcommand\HD{\hat D}\)
\( \newcommand\HE{\hat E}\)
\( \newcommand\HF{\hat F}\)
\( \newcommand\HG{\hat G}\)
\( \newcommand\HH{\hat H}\)
\( \newcommand\HI{\hat I}\)
\( \newcommand\HJ{\hat J}\)
\( \newcommand\HK{\hat K}\)
\( \newcommand\HL{\hat L}\)
\( \newcommand\HM{\hat M}\)
\( \newcommand\HN{\hat N}\)
\( \newcommand\HO{\hat O}\)
\( \newcommand\HP{\hat P}\)
\( \newcommand\HQ{\hat Q}\)
\( \newcommand\HR{\hat R}\)
\( \newcommand\HS{\hat S}\)
\( \newcommand\HT{\hat T}\)
\( \newcommand\HU{\hat U}\)
\( \newcommand\HV{\hat V}\)
\( \newcommand\HW{\hat W}\)
\( \newcommand\HX{\hat X}\)
\( \newcommand\HY{\hat Y}\)
\( \newcommand\HZ{\hat Z}\)
\( \newcommand\Halpha{\hat\alpha}\)
\( \newcommand\Hbeta{\hat\beta}\)
\( \newcommand\Hgamma{\hat\gamma}\)
\( \newcommand\Hdelta{\hat\delta}\)
\( \newcommand\Hepsilon{\hat\epsilon}\)
\( \newcommand\Hvarepsilon{\hat\varepsilon}\)
\( \newcommand\Hzeta{\hat\zeta}\)
\( \newcommand\Heta{\hat\eta}\)
\( \newcommand\Htheta{\hat\theta}\)
\( \newcommand\Hvartheta{\hat\vartheta}\)
\( \newcommand\Hiota{\hat\iota}\)
\( \newcommand\Hkappa{\hat\kappa}\)
\( \newcommand\Hlambda{\hat\lambda}\)
\( \newcommand\Hmu{\hat\mu}\)
\( \newcommand\Hnu{\hat\nu}\)
\( \newcommand\Hxi{\hat\xi}\)
\( \newcommand\Hom{\hat\omicron}\)
\( \newcommand\Hpi{\hat\pi}\)
\( \newcommand\Hvarpi{\hat\varpi}\)
\( \newcommand\Hrho{\hat\rho}\)
\( \newcommand\Hvarrho{\hat\varrho}\)
\( \newcommand\Hsigma{\hat\sigma}\)
\( \newcommand\Hvarsigma{\hat\varsigma}\)
\( \newcommand\Htau{\var\tau}\)
\( \newcommand\Hupsilon{\hat\upsilon}\)
\( \newcommand\Hphi{\hat\phi}\)
\( \newcommand\Hvarphi{\hat\varphi}\)
\( \newcommand\Hchi{\hat\chi}\)
\( \newcommand\Hxhi{\hat\xhi}\)
\( \newcommand\Hpsi{\hat\psi}\)
\( \newcommand\Homega{\hat\omega}\)
\( \newcommand\HGamma{\hat\Gamma}\)
\( \newcommand\HDelta{\hat\Delta}\)
\( \newcommand\HTheta{\hat\Theta}\)
\( \newcommand\HLambda{\hat\Lambda}\)
\( \newcommand\HXi{\hat\Xi}\)
\( \newcommand\HPi{\hat\Pi}\)
\( \newcommand\HSigma{\hat\Sigma}\)
\( \newcommand\HUps{\hat\Upsilon}\)
\( \newcommand\HPhi{\hat\Phi}\)
\( \newcommand\HPsi{\hat\Psi}\)
\( \newcommand\HOmega{\hat\Omega}\)
\( \newcommand\xhat{\hat\Bx}\)
\( \newcommand\yhat{\hat\By}\)
\( \newcommand\zhat{\hat\Bz}\)
\( \newcommand\ehat{\hat\Be}\)
\( \newcommand\khat{\hat\Bk}\)
\( \newcommand\nhat{\hat\Bn}\)
\( \newcommand\rhat{\hat\Br}\)
\( \newcommand\phihat{\hat\Bphi}\)
\( \newcommand\thetahat{\hat\Btheta}\)
\( \newcommand\MA{\mathbb A}\)
\( \newcommand\MB{\mathbb B}\)
\( \newcommand\MC{\mathbb C}\)
\( \newcommand\MD{\mathbb D}\)
\( \newcommand\ME{\mathbb E}\)
\( \newcommand\MF{\mathbb F}\)
\( \newcommand\MG{\mathbb G}\)
\( \newcommand\MH{\mathbb H}\)
\( \newcommand\MI{\mathbb I}\)
\( \newcommand\MJ{\mathbb J}\)
\( \newcommand\MK{\mathbb K}\)
\( \newcommand\ML{\mathbb L}\)
\( \newcommand\MM{\mathbb M}\)
\( \newcommand\MN{\mathbb N}\)
\( \newcommand\MO{\mathbb O}\)
\( \newcommand\MP{\mathbb P}\)
\( \newcommand\MQ{\mathbb Q}\)
\( \newcommand\MR{\mathbb R}\)
\( \newcommand\MS{\mathbb S}\)
\( \newcommand\MT{\mathbb T}\)
\( \newcommand\MU{\mathbb U}\)
\( \newcommand\MV{\mathbb V}\)
\( \newcommand\MW{\mathbb W}\)
\( \newcommand\MX{\mathbb X}\)
\( \newcommand\MY{\mathbb Y}\)
\( \newcommand\MZ{\mathbb Z}\)
\( \newcommand\CA{\mathcal A}\)
\( \newcommand\CB{\mathcal B}\)
\( \newcommand\CC{\mathcal C}\)
\( \newcommand\CD{\mathcal D}\)
\( \newcommand\CE{\mathcal E}\)
\( \newcommand\CF{\mathcal F}\)
\( \newcommand\CG{\mathcal G}\)
\( \newcommand\CH{\mathcal H}\)
\( \newcommand\CI{\mathcal I}\)
\( \newcommand\CJ{\mathcal J}\)
\( \newcommand\CK{\mathcal K}\)
\( \newcommand\CL{\mathcal L}\)
\( \newcommand\CM{\mathcal M}\)
\( \newcommand\CN{\mathcal N}\)
\( \newcommand\CO{\mathcal O}\)
\( \newcommand\CP{\mathcal P}\)
\( \newcommand\CQ{\mathcal Q}\)
\( \newcommand\CR{\mathcal R}\)
\( \newcommand\CS{\mathcal S}\)
\( \newcommand\CT{\mathcal T}\)
\( \newcommand\CU{\mathcal U}\)
\( \newcommand\CV{\mathcal V}\)
\( \newcommand\CW{\mathcal W}\)
\( \newcommand\CX{\mathcal X}\)
\( \newcommand\CY{\mathcal Y}\)
\( \newcommand\CZ{\mathcal Z}\)
\( \newcommand\Fa{\mathfrak a}\)
\( \newcommand\Fb{\mathfrak b}\)
\( \newcommand\Fc{\mathfrak c}\)
\( \newcommand\Fd{\mathfrak d}\)
\( \newcommand\Fe{\mathfrak e}\)
\( \newcommand\Ff{\mathfrak f}\)
\( \newcommand\Fg{\mathfrak g}\)
\( \newcommand\Fh{\mathfrak h}\)
\( \newcommand\Fi{\mathfrak i}\)
\( \newcommand\Fj{\mathfrak j}\)
\( \newcommand\Fk{\mathfrak k}\)
\( \newcommand\Fl{\mathfrak l}\)
\( \newcommand\Fm{\mathfrak m}\)
\( \newcommand\Fn{\mathfrak n}\)
\( \newcommand\Fo{\mathfrak o}\)
\( \newcommand\Fp{\mathfrak p}\)
\( \newcommand\Fq{\mathfrak q}\)
\( \newcommand\Fr{\mathfrak r}\)
\( \newcommand\Fs{\mathfrak s}\)
\( \newcommand\Ft{\mathfrak t}\)
\( \newcommand\Fu{\mathfrak u}\)
\( \newcommand\Fv{\mathfrak v}\)
\( \newcommand\Fw{\mathfrak w}\)
\( \newcommand\Fx{\mathfrak x}\)
\( \newcommand\Fy{\mathfrak y}\)
\( \newcommand\Fz{\mathfrak z}\)
\( \newcommand\FA{\mathfrak A}\)
\( \newcommand\FB{\mathfrak B}\)
\( \newcommand\FC{\mathfrak C}\)
\( \newcommand\FD{\mathfrak D}\)
\( \newcommand\FE{\mathfrak E}\)
\( \newcommand\FF{\mathfrak F}\)
\( \newcommand\FG{\mathfrak G}\)
\( \newcommand\FH{\mathfrak H}\)
\( \newcommand\FI{\mathfrak I}\)
\( \newcommand\FJ{\mathfrak J}\)
\( \newcommand\FK{\mathfrak K}\)
\( \newcommand\FL{\mathfrak L}\)
\( \newcommand\FM{\mathfrak M}\)
\( \newcommand\FN{\mathfrak N}\)
\( \newcommand\FO{\mathfrak O}\)
\( \newcommand\FP{\mathfrak P}\)
\( \newcommand\FQ{\mathfrak Q}\)
\( \newcommand\FR{\mathfrak R}\)
\( \newcommand\FS{\mathfrak S}\)
\( \newcommand\FT{\mathfrak T}\)
\( \newcommand\FU{\mathfrak U}\)
\( \newcommand\FV{\mathfrak V}\)
\( \newcommand\FW{\mathfrak W}\)
\( \newcommand\FX{\mathfrak X}\)
\( \newcommand\FY{\mathfrak Y}\)
\( \newcommand\FZ{\mathfrak Z}\)
\( \newcommand\Da{\dot a}\)
\( \newcommand\Db{\dot b}\)
\( \newcommand\Dc{\dot c}\)
\( \newcommand\Dd{\dot d}\)
\( \newcommand\De{\dot e}\)
\( \newcommand\Df{\dot f}\)
\( \newcommand\Dg{\dot g}\)
\( \newcommand\Dh{\dot h}\)
\( \newcommand\Di{\dot \imath}\)
\( \newcommand\Dj{\dot \jmath}\)
\( \newcommand\Dk{\dot k}\)
\( \newcommand\Dl{\dot l}\)
\( \newcommand\Dm{\dot m}\)
\( \newcommand\Dn{\dot n}\)
\( \newcommand\Do{\dot o}\)
\( \newcommand\Dp{\dot p}\)
\( \newcommand\Dq{\dot q}\)
\( \newcommand\Dr{\dot r}\)
\( \newcommand\Ds{\dot s}\)
\( \newcommand\Dt{\dot t}\)
\( \newcommand\Du{\dot u}\)
\( \newcommand\Dv{\dot v}\)
\( \newcommand\Dw{\dot w}\)
\( \newcommand\Dx{\dot x}\)
\( \newcommand\Dy{\dot y}\)
\( \newcommand\Dz{\dot z}\)
\( \newcommand\DA{\dot A}\)
\( \newcommand\DB{\dot B}\)
\( \newcommand\DC{\dot C}\)
\( \newcommand\DD{\dot D}\)
\( \newcommand\DE{\dot E}\)
\( \newcommand\DF{\dot F}\)
\( \newcommand\DG{\dot G}\)
\( \newcommand\DH{\dot H}\)
\( \newcommand\DI{\dot I}\)
\( \newcommand\DJ{\dot J}\)
\( \newcommand\DK{\dot K}\)
\( \newcommand\DL{\dot L}\)
\( \newcommand\DM{\dot M}\)
\( \newcommand\DN{\dot N}\)
\( \newcommand\DO{\dot O}\)
\( \newcommand\DP{\dot P}\)
\( \newcommand\DQ{\dot Q}\)
\( \newcommand\DR{\dot R}\)
\( \newcommand\DS{\dot S}\)
\( \newcommand\DT{\dot T}\)
\( \newcommand\DU{\dot U}\)
\( \newcommand\DV{\dot V}\)
\( \newcommand\DW{\dot W}\)
\( \newcommand\DX{\dot X}\)
\( \newcommand\DY{\dot Y}\)
\( \newcommand\DZ{\dot Z}\)
\( \newcommand\Dalpha
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dbeta
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dgamma
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Ddelta
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Depsilon
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dvarepsilon
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dzeta
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Deta
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dtheta
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dvartheta
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Diota
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dkappa
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dlambda
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dmu{\dot\mu}\)
\( \newcommand\Dnu{\dot\nu}\)
\( \newcommand\Dxi{\dot\xi}\)
\( \newcommand\Dom{\dot\omicron}\)
\( \newcommand\Dpi{\dot\pi}\)
\( \newcommand\Dvarpi
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Drho{\dot\rho}\)
\( \newcommand\Dvarrho{\dot\varrho}\)
\( \newcommand\Dsigma{\dot\sigma}\)
\( \newcommand\Dvarsigma{\dot\varsigma}\)
\( \newcommand\Dtau{\var\tau}\)
\( \newcommand\Dupsilon{\dot\upsilon}\)
\( \newcommand\Dphi{\dot\phi}\)
\( \newcommand\Dvarphi{\dot\varphi}\)
\( \newcommand\Dchi{\dot\chi}\)
\( \newcommand\Dpsi{\dot\psi}\)
\( \newcommand\Domega{\dot\omega}\)
\( \newcommand\DGamma
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\DDelta
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\DTheta
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\DLambda{\dot\Lambda}\)
\( \newcommand\DXi{\dot\Xi}\)
\( \newcommand\DPi{\dot\Pi}\)
\( \newcommand\DSigma{\dot\Sigma}\)
\( \newcommand\DUps{\dot\Upsilon}\)
\( \newcommand\DPhi{\dot\Phi}\)
\( \newcommand\DPsi{\dot\Psi}\)
\( \newcommand\DOmega{\dot\Omega}\)
\( \newcommand\Va{\vec a}\)
\( \newcommand\Vb{\vec b}\)
\( \newcommand\Vc{\vec c}\)
\( \newcommand\Vd{\vec d}\)
\( \newcommand\Ve{\vec e}\)
\( \newcommand\Vf{\vec f}\)
\( \newcommand\Vg{\vec g}\)
\( \newcommand\Vh{\vec h}\)
\( \newcommand\Vi{\vec \imath}\)
\( \newcommand\Vj{\vec \jmath}\)
\( \newcommand\Vk{\vec k}\)
\( \newcommand\Vl{\vec l}\)
\( \newcommand\Vm{\vec m}\)
\( \newcommand\Vn{\vec n}\)
\( \newcommand\Vo{\vec o}\)
\( \newcommand\Vp{\vec p}\)
\( \newcommand\Vq{\vec q}\)
\( \newcommand\Vr{\vec r}\)
\( \newcommand\Vs{\vec s}\)
\( \newcommand\Vt{\vec t}\)
\( \newcommand\Vu{\vec u}\)
\( \newcommand\Vv{\vec v}\)
\( \newcommand\Vw{\vec w}\)
\( \newcommand\Vx{\vec x}\)
\( \newcommand\Vy{\vec y}\)
\( \newcommand\Vz{\vec z}\)
\( \newcommand\VA{\vec A}\)
\( \newcommand\VB{\vec B}\)
\( \newcommand\VC{\vec C}\)
\( \newcommand\VD{\vec D}\)
\( \newcommand\VE{\vec E}\)
\( \newcommand\VF{\vec F}\)
\( \newcommand\VG{\vec G}\)
\( \newcommand\VH{\vec H}\)
\( \newcommand\VI{\vec I}\)
\( \newcommand\VJ{\vec J}\)
\( \newcommand\VK{\vec K}\)
\( \newcommand\VL{\vec L}\)
\( \newcommand\VM{\vec M}\)
\( \newcommand\VN{\vec N}\)
\( \newcommand\VO{\vec O}\)
\( \newcommand\VP{\vec P}\)
\( \newcommand\VQ{\vec Q}\)
\( \newcommand\VR{\vec R}\)
\( \newcommand\VS{\vec S}\)
\( \newcommand\VT{\vec T}\)
\( \newcommand\VU{\vec U}\)
\( \newcommand\VV{\vec V}\)
\( \newcommand\VW{\vec W}\)
\( \newcommand\VX{\vec X}\)
\( \newcommand\VY{\vec Y}\)
\( \newcommand\VZ{\vec Z}\)
\( \newcommand\Valpha{\vec\alpha}\)
\( \newcommand\Vbeta{\vec\beta}\)
\( \newcommand\Vgamma{\vec\gamma}\)
\( \newcommand\Vdelta{\vec\delta}\)
\( \newcommand\Vepsilon{\vec\epsilon}\)
\( \newcommand\Vvarepsilon{\vec\varepsilon}\)
\( \newcommand\Vzeta{\vec\zeta}\)
\( \newcommand\Veta{\vec\eta}\)
\( \newcommand\Vtheta{\vec\theta}\)
\( \newcommand\Vvartheta{\vec\vartheta}\)
\( \newcommand\Viota{\vec\iota}\)
\( \newcommand\Vkappa{\vec\kappa}\)
\( \newcommand\Vlambda{\vec\lambda}\)
\( \newcommand\Vmu
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vnu
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vxi
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vom
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vpi
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarpi
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vrho
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarrho
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vsigma
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarsigma
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vtau
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vupsilon
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vphi
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarphi
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vchi
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vpsi
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vomega
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\VGamma
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\VDelta
Callstack:
at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
\( \newcommand\BB{\mib B}\)
\( \newcommand\BC{\mib C}\)
\( \newcommand\BD{\mib D}\)
\( \newcommand\BE{\mib E}\)
\( \newcommand\BF{\mib F}\)
\( \newcommand\BG{\mib G}\)
\( \newcommand\BH{\mib H}\)
\( \newcommand\BI{\mib I}}\)
\( \newcommand\BJ{\mib J}\)
\( \newcommand\BK{\mib K}\)
\( \newcommand\BL{\mib L}\)
\( \newcommand\BM{\mib M}\)
\( \newcommand\BN{\mib N}\)
\( \newcommand\BO{\mib O}\)
\( \newcommand\BP{\mib P}\)
\( \newcommand\BQ{\mib Q}\)
\( \newcommand\BR{\mib R}\)
\( \newcommand\BS{\mib S}\)
\( \newcommand\BT{\mib T}\)
\( \newcommand\BU{\mib U}\)
\( \newcommand\BV{\mib V}\)
\( \newcommand\BW{\mib W}\)
\( \newcommand\BX{\mib X}\)
\( \newcommand\BY{\mib Y}\)
\( \newcommand\BZ{\mib Z}\)
\( \newcommand\Ba{\mib a}\)
\( \newcommand\Bb{\mib b}\)
\( \newcommand\Bc{\mib c}\)
\( \newcommand\Bd{\mib d}\)
\( \newcommand\Be{\mib e}\)
\( \newcommand\Bf{\mib f}\)
\( \newcommand\Bg{\mib g}\)
\( \newcommand\Bh{\mib h}\)
\( \newcommand\Bi{\mib i}\)
\( \newcommand\Bj{\mib j}\)
\( \newcommand\Bk{\mib k}\)
\( \newcommand\Bl{\mib l}\)
\( \newcommand\Bm{\mib m}\)
\( \newcommand\Bn{\mib n}\)
\( \newcommand\Bo{\mib o}\)
\( \newcommand\Bp{\mib p}\)
\( \newcommand\Bq{\mib q}\)
\( \newcommand\Br{\mib r}\)
\( \newcommand\Bs{\mib s}\)
\( \newcommand\Bt{\mib t}\)
\( \newcommand\Bu{\mib u}\)
\( \newcommand\Bv{\mib v}\)
\( \newcommand\Bw{\mib w}\)
\( \newcommand\Bx{\mib x}\)
\( \newcommand\By{\mib y}\)
\( \newcommand\Bz{\mib z}\)\)
\( \newcommand\vrh{\varrho}\)
\( \newcommand\vsig{\varsigma}\)
\( \newcommand\ups{\upsilon}\)
\( \newcommand\eps{\epsilon}\)
\( \newcommand\ve{\varepsilon}\)
\( \newcommand\vth{\vartheta}\)
\( \newcommand\vphi{\varphi}\)
\( \newcommand\xhi{\chi}\)
\( \newcommand\Ups{\Upsilon}\)
\( \newcommand\Balpha{\mib\alpha}\)
\( \newcommand\Bbeta{\mib\beta}\)
\( \newcommand\Bgamma{\mib\gamma}\)
\( \newcommand\Bdelta{\mib\delta}\)
\( \newcommand\Beps{\mib\epsilon}\)
\( \newcommand\Bve{\mib\varepsilon}\)
\( \newcommand\Bzeta{\mib\zeta}\)
\( \newcommand\Beta{\mib\eta}\)
\( \newcommand\Btheta{\mib\theta}\)
\( \newcommand\Bvth{\mib\vartheta}\)
\( \newcommand\Biota{\mib\iota}\)
\( \newcommand\Bkappa{\mib\kappa}\)
\( \newcommand\Blambda{\mib\lambda}\)
\( \newcommand\Bmu{\mib\mu}\)
\( \newcommand\Bnu{\mib\nu}\)
\( \newcommand\Bxi{\mib\xi}\)
\( \newcommand\Bom{\mib\omicron}\)
\( \newcommand\Bpi{\mib\pi}\)
\( \newcommand\Bvarpi{\mib\varpi}\)
\( \newcommand\Brho{\mib\rho}\)
\( \newcommand\Bvrh{\mib\varrho}\)
\( \newcommand\Bsigma{\mib\sigma}\)
\( \newcommand\Bvsig{\mib\varsigma}\)
\( \newcommand\Btau{\mib\tau}\)
\( \newcommand\Bups{\mib\upsilon}\)
\( \newcommand\Bphi{\mib\phi}\)
\( \newcommand\Bvphi{\mib\vphi}\)
\( \newcommand\Bchi{\mib\chi}\)
\( \newcommand\Bpsi{\mib\psi}\)
\( \newcommand\Bomega{\mib\omega}\)
\( \newcommand\BGamma{\mib\Gamma}\)
\( \newcommand\BDelta{\mib\Delta}\)
\( \newcommand\BTheta{\mib\Theta}\)
\( \newcommand\BLambda{\mib\Lambda}\)
\( \newcommand\BXi{\mib\Xi}\)
\( \newcommand\BPi{\mib\Pi}\)
\( \newcommand\BSigma{\mib\Sigma}\)
\( \newcommand\BUps{\mib\Upsilon}\)
\( \newcommand\BPhi{\mib\Phi}\)
\( \newcommand\BPsi{\mib\Psi}\)
\( \newcommand\BOmega{\mib\Omega}\)
\( \newcommand\Bxhi{\raise.35ex\hbox{$\Bchi$}}\)
\( \newcommand\RGamma{ \Gamma}\)
\( \newcommand\RDelta{ \Delta}\)
\( \newcommand\RTheta{ \Theta}\)
\( \newcommand\RLambda{ \Lambda}\)
\( \newcommand\RXi{ \Xi}\)
\( \newcommand\RPi{ \Pi}\)
\( \newcommand\RSigma{ \Sigma}\)
\( \newcommand\RUps{ \Upsilon}\)
\( \newcommand\RPhi{ \Phi}\)
\( \newcommand\RPsi{ \Psi}\)
\( \newcommand\ROmega{ \Omega}\)
\( \newcommand\RA{ A}\)
\( \newcommand\RB{ B}\)
\( \newcommand\RC{ C}\)
\( \newcommand\RD{ D}\)
\( \newcommand\RE{ E}\)
\( \newcommand\RF{ F}\)
\( \newcommand\RG{ G}\)
\( \newcommand\RH{ H}\)
\( \newcommand\RI{ I}\)
\( \newcommand\RJ{ J}\)
\( \newcommand\RK{ K}\)
\( \newcommand\RL{ L}\)
\( \newcommand { M}\)
\( \newcommand\RN{ N}\)
\( \newcommand\RO{ O}\)
\( \newcommand\RP{ P}\)
\( \newcommand\RQ{ Q}\)
\( \newcommand\RR{ R}\)
\( \newcommand\RS{ S}\)
\( \newcommand\RT{ T}\)
\( \newcommand\RU{ U}\)
\( \newcommand\RV{ V}\)
\( \newcommand\RW{ W}\)
\( \newcommand\RX{ X}\)
\( \newcommand\RY{ Y}\)
\( \newcommand\RZ{ Z}\)
\( \newcommand\Ra{ a}\)
\( \newcommand\Rb{ b}\)
\( \newcommand\Rc{ c}\)
\( \newcommand\Rd{ d}\)
\( \newcommand\Re{ e}\)
\( \newcommand\Rf{ f}\)
\( \newcommand\Rg{ g}\)
\( \newcommand\Rh{ h}\)
\( \newcommand\Ri{ i}\)
\( \newcommand\Rj{ j}\)
\( \newcommand\Rk{ k}\)
\( \newcommand\Rl{ l}\)
\( \newcommand { m}\)
\( \newcommand\Rn{ n}\)
\( \newcommand\Ro{ o}\)
\( \newcommand\Rp{ p}\)
\( \newcommand\Rq{ q}\)
\( \newcommand\Rr{ r}\)
\( \newcommand\Rs{ s}\)
\( \newcommand\Rt{ t}\)
\( \newcommand\Ru{ u}\)
\( \newcommand\Rv{ v}\)
\( \newcommand\Rw{ w}\)
\( \newcommand\Rx{ x}\)
\( \newcommand\Ry{ y}\)
\( \newcommand\Rz{ z}\)
\( \newcommand\BBA{\boldsymbol\RA}\)
\( \newcommand\BBB{\boldsymbol\RB}\)
\( \newcommand\BBC{\boldsymbol\RC}\)
\( \newcommand\BBD{\boldsymbol\RD}\)
\( \newcommand\BBE{\boldsymbol\RE}\)
\( \newcommand\BBF{\boldsymbol\RF}\)
\( \newcommand\BBG{\boldsymbol\RG}\)
\( \newcommand\BBH{\boldsymbol\RH}\)
\( \newcommand\BBI{\boldsymbol\RI}\)
\( \newcommand\BBJ{\boldsymbol\RJ}\)
\( \newcommand\BBK{\boldsymbol\RK}\)
\( \newcommand\BBL{\boldsymbol\RL}\)
\( \newcommand\BBM{\boldsymbol }\)
\( \newcommand\BBN{\boldsymbol\RN}\)
\( \newcommand\BBO{\boldsymbol\RO}\)
\( \newcommand\BBP{\boldsymbol\RP}\)
\( \newcommand\BBQ{\boldsymbol\RQ}\)
\( \newcommand\BBR{\boldsymbol\RR}\)
\( \newcommand\BBS{\boldsymbol\RS}\)
\( \newcommand\BBT{\boldsymbol\RT}\)
\( \newcommand\BBU{\boldsymbol\RU}\)
\( \newcommand\BBV{\boldsymbol\RV}\)
\( \newcommand\BBW{\boldsymbol\RW}\)
\( \newcommand\BBX{\boldsymbol\RX}\)
\( \newcommand\BBY{\boldsymbol\RY}\)
\( \newcommand\BBZ{\boldsymbol\RZ}\)
\( \newcommand\BBa{\boldsymbol\Ra}\)
\( \newcommand\BBb{\boldsymbol\Rb}\)
\( \newcommand\BBc{\boldsymbol\Rc}\)
\( \newcommand\BBd{\boldsymbol\Rd}\)
\( \newcommand\BBe{\boldsymbol\Re}\)
\( \newcommand\BBf{\boldsymbol\Rf}\)
\( \newcommand\BBg{\boldsymbol\Rg}\)
\( \newcommand\BBh{\boldsymbol\Rh}\}\)
\( \newcommand\BBi{\boldsymbol\Ri}\)
\( \newcommand\BBj{\boldsymbol\Rj}\)
\( \newcommand\BBk{\boldsymbol\Rk}\)
\( \newcommand\BBl{boldsymbol\Rl}\)
\( \newcommand\BBm{\boldsymbol }\)
\( \newcommand\BBn{\boldsymbol\Rn}\)
\( \newcommand\BBo{\boldsymbol\Ro}\)
\( \newcommand\BBp{\boldsymbol\Rp}\)
\( \newcommand\BBq{\boldsymbol\Rq}\)
\( \newcommand\BBr{\boldsymbol\Rr}\)
\( \newcommand\BBs{\boldsymbol\Rs}\)
\( \newcommand\BBt{\boldsymbol\Rt}\)
\( \newcommand\BBu{\boldsymbol\Ru}\)
\( \newcommand\BBv{\boldsymbol\Rv}\)
\( \newcommand\BBw{\boldsymbol\Rw}\)
\( \newcommand\BBx{\boldsymbol\Rx}\)
\( \newcommand\BBy{\boldsymbol\Ry}\)
\( \newcommand\BBz{\boldsymbol\Rz}\)
\( \newcommand\tcb{\textcolor{blue}\)
\( \newcommand\tcr{\textcolor{red}\)
\( \newcommand\bnabla{\boldsymbol{\nabla}}\)
\( \newcommand\Bell{\boldsymbol\ell}\)
\( \newcommand\dbar{\,{\mathchar'26\mkern-12mu d}} \)
\( \newcommand\ns{^\vphantom{*}}\)
\( \newcommand\uar{\uparrow}\)
\( \newcommand\dar{\downarrow}\)
\( \newcommand\impi{\int\limits_{-\infty}^{\infty}\!\!}\)
\( \newcommand\izpi{\int\limits_{0}^{\infty}\!\!}\)
\( \newcommand\etc{\it etc.\/}\)
\( \newcommand\etal{\it et al.\/}\)
\( \newcommand\opcit{\it op. cit.\/}\)
\( \newcommand\ie{\it i.e.\/}\)
\( \newcommand\Ie{\it I.e.\/}\)
\( \newcommand\viz{\it viz.\/}\)
\( \newcommand\eg{\it e.g.\/}\)
\( \newcommand\Eg{\it E.g.\/}\)
\( \newcommand\dbar{\,{\mathchar'26\mkern-12mu d}} \)
\( \def\sss#1{\scriptscriptstyle #1}\)
\( \def\ss#1{\scriptstyle #1}\)
\( \def\ssr#1{\scriptstyle #1}\)
\( \def\ssf#1{\scriptstyle #1}\)
\( \newcommand\NA{N_{\ssr{\!A}}}\)
\( \newcommand\lala{\langle\!\langle}\)
\( \newcommand\rara{\rangle\!\rangle}\)
\( \newcommand\blan{\big\langle}\)
\( \newcommand\bran{\big\rangle}\)
\( \newcommand\Blan{\Big\langle}\)
\( \newcommand\Bran{\Big\rangle}\)
\( \newcommand\intl{\int\limits}\)
\( \newcommand\half{\frac{1}{2}}\)
\( \newcommand\third{\frac{1}{3}}\)
\( \newcommand\fourth{\frac{1}{4}}\)
\( \newcommand\eighth{\frac{1}{8}}\)
\( \newcommand\uar{\uparrow}\)
\( \newcommand\dar{\downarrow}\)
\( \newcommand\undertext#1{$\underline{\hbox{#1}}$}\)
\( \newcommand\Tra{\mathop{\textsf{Tr}}\,}\)
\( \newcommand\det{\mathop{\textsf{det}}\,}\)
\( \def\tket#1{| #1 \rangle}\)
\( \def\tbra#1{\langle #1|}\)
\( \def\tbraket#1#2{\langle #1 | #2 \rangle}\)
\( \def\texpect#1#2#3{\langle #1 | #2 | #3 \rangle}\)
\( \def\sket#1{| \, #1 \, \rangle}\)
\( \def\sbra#1{\langle \, #1 \, |}\)
\( \def\sbraket#1#2{\langle \, #1 \, | \, #2 \, \rangle}\)
\( \def\sexpect#1#2#3{\langle \, #1 \, | \, #2 \, | \, #3 \, \rangle}\)
\(\def\ket#1{\big| \, #1\, \big\rangle}\)
\( \def\bra#1{\big\langle \, #1 \, \big|}\)
\( \def\braket#1#2{\big\langle \, #1\, \big| \,#2 \,\big\rangle}\)
\( \def\expect#1#2#3{\big\langle\, #1\, \big|\, #2\, \big| \,#3\, \big\rangle}\)
\( \newcommand\pz{\partial}\)
\( \newcommand\pzb{\bar{\partial}}\)
\( \newcommand\svph{\vphantom{\int}}\)
\( \newcommand\vph{\vphantom{\sum_i}}\)
\( \newcommand\bvph{\vphantom{\sum_N^N}}\)
\( \newcommand\nd{^{\vphantom{\dagger}}}\)
\( \newcommand\ns{^{\vphantom{*}}}\)
\( \newcommand\yd{^\dagger}\)
\( \newcommand\zb{\bar z}\)
\( \newcommand\zdot{\dot z}\)
\( \newcommand\zbdot{\dot{\bar z}}\)
\( \newcommand\kB{k_{\sss{B}}}\)
\( \newcommand\kT{k_{\sss{B}}T}\)
\( \newcommand\gtau{g_\tau}\)
\( \newcommand\Htil{\tilde H}\)
\( \newcommand\pairo{(\phi\nd_0,J\nd_0)}\)
\( \newcommand\pairm{(\phi\nd_0,J)}\)
\( \newcommand\pairob{(\Bphi\nd_0,\BJ\nd_0)}\)
\( \newcommand\pairmb{(\Bphi\nd_0,\BJ)}\)
\( \newcommand\pair{(\phi,J)}\)
\( \newcommand\Hz{H\nd_0}\)
\( \newcommand\Ho{H\nd_1}\)
\( \newcommand\Htz{\Htil\nd_0}\)
\( \newcommand\Hto{\Htil\nd_1}\)
\( \newcommand\oc{\omega_\Rc}\)
\(\newcommand \gtwid{\approx}\)
\( \newcommand\index{\textsf{ind}}\)
\( \newcommand\csch{\,{ csch\,}}\)
\( \newcommand\ctnh{\,{ ctnh\,}}\)
\( \newcommand\ctn{\,{ ctn\,}}\)
\( \newcommand\sgn{\,{ sgn\,}}\)
\( \def\tmapright#1{\xrightarrow \limits^{#1}}\)
\( \def\bmapright#1{\xrightarrow\limits_{#1}}\)
\( \newcommand\hfb{\hfill\break}\)
\( \newcommand\Rep{\textsf{Re}\,}\)
\( \newcommand\Imp{\textsf{Im}\,}\)
\( \newcommand\ncdot{\!\cdot\!}\)
\( \def\tmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits^{#1}}\ }\)
\( \def\bmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits_{#1}}\ }\)
\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)
\( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
\( \def\qabc#1#2#3{\pz^2\! #1\over\pz #2\,\pz #3}\)
\( \def\rabc#1#2#3#4{(\pz #1,\pz #2)\over (\pz #3,\pz #4)}\)
\( \newcommand\subA{\ns_\ssr{A}}\)
\( \newcommand\subB{\ns_\ssr{B}}\)
\( \newcommand\subC{\ns_\ssr{C}}\)
\( \newcommand\subD{\ns_\ssr{D}}\)
\( \newcommand\subAB{\ns_\ssr{AB}}\)
\( \newcommand\subBC{\ns_\ssr{BC}}\)
\( \newcommand\subCD{\ns_\ssr{CD}}\)
\( \newcommand\subDA{\ns_\ssr{DA}}\)
\( \def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }\)
\( \def\enth#1{\RDelta {\textsf H}^0_\Rf[{ #1}]}\)
\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
\( \newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}\)
\( \newcommand\longleftharpoondown{\leftharpoondown\joinrel\relbar}\)
\( \newcommand\cds{\,\bullet\,}\)
\( \newcommand\ccs{\,\circ\,}\)
\( \newcommand\nsub{_{\vphantom{\dagger}}}\)
\( \newcommand\rhohat{\hat\rho}\)
\( \newcommand\vrhhat{\hat\vrh}\)
\( \newcommand\impi{\int\limits_{-\infty}^\infty\!\!\!}\)
\( \newcommand\brangle{\big\rangle}\)
\( \newcommand\blangle{\big\langle}\)
\( \newcommand\vet{\tilde\ve}\)
\( \newcommand\zbar{\bar z}\)
\( \newcommand\ftil{\tilde f}\)
\( \newcommand\XBE{\RXi\ns_\ssr{BE}}\)
\( \newcommand\XFD{\RXi\ns_\ssr{FD}}\)
\( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
\( \newcommand\OFD{\Omega\ns_\ssr{FD}}\)
\( \newcommand\veF{\ve\ns_\RF}\)
\( \newcommand\kF{k\ns_\RF}\)
\( \newcommand\kFu{k\ns_{\RF\uar}}\)
\( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
\( \newcommand\muB{\mu\ns_\ssr{B}}\)
\( \newcommand\mutB{\tilde\mu}\ns_\ssr{B}\)
\( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
\( \newcommand\xhihOZ
Callstack:
at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\jhz{\HJ(0)}\)
\( \newcommand\nda{\nd_\alpha}\)
\( \newcommand\ndap{\nd_{\alpha'}}\)
\( \newcommand\labar
Callstack:
at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1
at template()
at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05:_Noninteracting_Quantum_Systems/5.10:_Appendix_II-_Ideal_Bose_Gas_Condensation), /content/body/p[1]/span, line 1, column 23
\( \newcommand\msa{m\ns_\ssr{A}}\)
\( \newcommand\msb{m\ns_\ssr{B}}\)
\( \newcommand\mss{m\ns_\Rs}\)
\( \newcommand\HBx{\hat\Bx}\)
\( \newcommand\HBy{\hat\By}\)
\( \newcommand\HBz{\hat\Bz}\)
\( \newcommand\thm{\theta\ns_m}\)
\( \newcommand\thp{\theta\ns_\phi}\)
\( \newcommand\mtil{\widetilde m}\)
\( \newcommand\phitil{\widetilde\phi}\)
\( \newcommand\delf{\delta\! f}\)
\( \newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}\)
\( \newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}\)
\( \newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}\)
\( \newcommand\vbar{\bar v}\)
\( \newcommand\BCE{\mbox{\boldmath{$\CE$}}\!}\)
\( \newcommand\BCR{\mbox{\boldmath{$\CR$}}\!}\)
\( \newcommand\gla{g\nd_{\RLambda\nd}}\)
\( \newcommand\TA{T\ns_\ssr{A}}\)
\( \newcommand\TB{T\ns_\ssr{B}}\)
\( \newcommand\ncdot{\!\cdot\!}\)
\( \newcommand\NS{N\ns_{\textsf S}}\)
We begin with the grand canonical Hamiltonian \(K=H-\mu N\) for the ideal Bose gas, \[K=\sum_\Bk (\ve\ns_\Bk-\mu)\,b\yd_\Bk b\nd_\Bk - \sqrt{N}\sum_\Bk\big(\nu\ns_\Bk \,b\yd_\Bk + {\bar\nu}\nd_\Bk\, b\nd_\Bk\big)\quad.\] Here \(b\yd_\Bk\) is the creation operator for a boson in a state of wavevector \(\Bk\), hence \(\big[b\nd_\Bk\,,\,b\yd_{\Bk'}\big]=\delta\ns_{\Bk\Bk'}\). The dispersion relation is given by the function \(\ve\ns_\Bk\), which is the energy of a particle with wavevector \(\Bk\). We must have \(\ve\ns_\Bk-\mu\ge 0\) for all \(\Bk\), lest the spectrum of \(K\) be unbounded from below. The fields \(\{\nu\ns_\Bk,{\bar\nu}\ns_\Bk\}\) break a global \(\SO(2)\) symmetry.
Students who have not taken a course in solid state physics can skip the following paragraph, and be aware that \(N=V/v\ns_0\) is the total volume of the system in units of a fundamental "unit cell" volume. The thermodynamic limit is then \(N\to\infty\). Note that \(N\) is not the boson particle number, which we’ll call \(N\ns_b\).
Solid state physics boilerplate : We presume a setting in which the real space Hamiltonian is defined by some boson hopping model on a Bravais lattice. The wavevectors \(\Bk\) are then restricted to the first Brillouin zone, \({\hat\ROmega}\), and assuming periodic boundary conditions are quantized according to the condition \(\exp\!\big(i N_l\, \Bk\,\cdot\Ba\ns_l\big) = 1\) for all \(l\in\{1,\ldots,d\}\), where \(\Ba\ns_l\) is the \(l^{th}\) fundamental direct lattice vector and \(N\ns_l\) is the size of the system in the \(\Ba\ns_l\) direction; \(d\) is the dimension of space. The total number of unit cells is \(N\equiv \prod_l N\ns_l\) . Thus, quantization entails \(\Bk=\sum_l (2\pi n\ns_l/N\ns_l)\,\Bb\ns_l\) , where \(\Bb\ns_l\) is the \(l^{th}\) elementary reciprocal lattice vector (\(\Ba\ns_l\cdot\Bb\ns_{l'}=2\pi\delta\ns_{ll'}\)) and \(n\ns_l\) ranges over \(N\ns_l\) distinct integers such that the allowed \(\Bk\) points form a discrete approximation to \({\hat\ROmega}\) .
To solve, we first shift the boson creation and annihilation operators, writing \[K=\sum_\Bk (\ve\ns_\Bk-\mu)\,\beta\yd_\Bk\beta\nd_\Bk - N\sum_\Bk {|\nu\nd_\Bk|^2\over \ve\ns_\Bk-\mu}\quad,\] where \[\beta\nd_\Bk= b\nd_\Bk - {\sqrt{N}\,\nu\nd_\Bk\over \ve\ns_\Bk-\mu}\qquad,\qquad \beta\yd_\Bk= b\yd_\Bk - {\sqrt{N}\,{\bar\nu}_\Bk\over \ve\ns_\Bk-\mu}\quad.\] Note that \(\big[\beta\nd_\Bk\,,\,\beta\yd_{\Bk'}\big]=\delta\nd_{\Bk\Bk'}\) so the above transformation is canonical. The Landau free energy \(\Omega=-\kT\ln\Xi\) , where \(\Xi=\Tra\,e^{-K/k\ns_\RB T}\), is given by \[\Omega=N\kT\!\!\impi d\ve\>g(\ve)\,\ln\big(1-e^{(\mu-\ve)/k\ns_\Rb T}\big) - N\sum_\Bk { |\nu\nd_\Bk|^2\over \ve\nd_\Bk - \mu} \quad,\] where \(g(\ve)\) is the density of energy states per unit cell, \[g(\ve)={1\over N}\sum_\Bk\delta\big(\ve-\ve\ns_\Bk\big)\bmapright{N\to\infty}\int\limits_{\hat\ROmega}\!\!{ d^d\!k\over (2\pi)^d}\>\delta\big(\ve-\ve\ns_\Bk\big) \quad.\] Note that \[\psi\ns_\Bk\equiv {1\over\sqrt{N}}\,\blangle b\nd_\Bk \brangle = -{1\over N}\,{\pz\Omega\over\pz{\bar\nu}\nd_\Bk} = {\nu\nd_\Bk\over\ve\ns_\Bk-\mu}\quad.\] In the condensed phase, \(\psi\ns_\Bk\) is nonzero.
The Landau free energy (grand potential) is a function \(\Omega(T,N,\mu,\nu,{\bar\nu})\). We now make a Legendre transformation, \[Y(T,N,\mu,\psi,{\bar\psi})=\Omega(T,N,\mu,\nu,{\bar\nu}) + N\sum_\Bk\big(\nu\ns_\Bk{\bar\psi}\ns_\Bk + {\bar\nu}\ns_\Bk\psi\ns_\Bk\big)\quad.\] Note that \[{\pz Y\over\pz{\bar\nu}\ns_\Bk}={\pz\Omega\over\pz{\bar\nu}\ns_\Bk} + N\psi\ns_\Bk=0\quad,\] by the definition of \(\psi\ns_\Bk\). Similarly, \(\pz Y/\pz\nu\ns_\Bk=0\). We now have \[Y(T,N,\mu,\psi,{\bar\psi})=N\kT\!\!\impi d\ve\>g(\ve)\,\ln\big(1-e^{(\mu-\ve)/k\ns_\Rb T}\big) + N\sum_\Bk(\ve\ns_\Bk-\mu)\,|\psi\nd_\Bk|^2\quad.\] Therefore, the boson particle number per unit cell is given by the dimensionless density, \[n={N\ns_b\over N}=-{1\over N}{\pz Y\over\pz\mu} = \sum_\Bk |\psi\nd_\Bk|^2 +\! \impi d\ve\>{g(\ve)\over e^{(\ve-\mu)/k\nd_\RB T} - 1 } \quad,\] and the condensate amplitude at wavevector \(\Bk\) is \[\nu\ns_\Bk = {1\over N}{\pz Y\over\pz{\bar\psi}\ns_\Bk\vphantom{A^B}} = (\ve\ns_\Bk-\mu)\,\psi\ns_\Bk\quad.\]
Recall that \(\nu\ns_\Bk\) acts as an external field. Let the dispersion \(\ve\ns_\Bk\) be minimized at \(\Bk=\BK\) . Without loss of generality, we may assume this minimum value is \(\ve\ns_\BK=0\) . We see that if \(\nu\ns_\Bk=0\) then one of two must be true:
- \(\psi\ns_\Bk=0\) for all \(\Bk\)
- \(\mu=\ve\ns_\BK\) , in which case \(\psi\ns_\BK\) can be nonzero.
Thus, for \(\nu={\bar\nu}=0\) and \(\mu>0\), we have the usual equation of state, \[n(T,\mu)=\! \impi d\ve\>{g(\ve)\over e^{(\ve-\mu)/k\nd_\RB T} - 1 } \quad, \label{GDE}\] which relates the intensive variables \(n\), \(T\), and \(\mu\). When \(\mu=0\), the equation of state becomes \[n(T,\mu=0) = \stackrel{n\ns_0}{\overbrace{\sum_\BK |\psi\nd_\BK|^2}} + \! \stackrel{n\ns_>(T)}{\overbrace{\impi d\ve\>{g(\ve)\over e^{\ve/k\nd_\RB T} - 1 }}} \quad, \label{condeqn}\] where now the sum is over only those \(\BK\) for which \(\ve\ns_\BK=0\) . Typically this set has only one member, \(\BK=0\), but it is quite possible, due to symmetry reasons, that there are more such \(\BK\) values. This last equation of state is one which relates the intensive variables \(n\), \(T\), and \(n\ns_0\) , where \[n\ns_0=\sum_\BK |\psi\nd_\BK|^2\] is the dimensionless condensate density. If the integral \(n\ns_>(T)\) in Equation [condeqn] is finite, then for \(n>n\ns_0(T)\) we must have \(n\ns_0>0\). Note that, for any \(T\), \(n\ns_>(T)\) diverges logarithmically whenever \(g(0)\) is finite. This means that Equation [GDE] can always be inverted to yield a finite \(\mu(n,T)\), no matter how large the value of \(n\), in which case there is no condensation and \(n\ns_0=0\). If \(g(\ve)\propto\ve^\alpha\) with \(\alpha>0\), the integral converges and \(n\ns_>(T)\) is finite and monotonically increasing for all \(T\). Thus, for fixed dimensionless number \(n\), there will be a critical temperature \(T\ns_\Rc\) for which \(n=n\ns_>(T\ns_\Rc)\). For \(T<T\ns_\Rc\) , Equation [GDE] has no solution for any \(\mu\) and we must appeal to Equation [condeqn]. The condensate density, given by \(n\ns_0(n,T)=n-n\ns_>(T)\) , is then finite for \(T<T\ns_\Rc\) , and vanishes for \(T\ge T\ns_\Rc\) .
In the condensed phase, the phase of the order parameter \(\psi\) inherits its phase from the external field \(\nu\), which is taken to zero, in the same way the magnetization in the symmetry-broken phase of an Ising ferromagnet inherits its direction from an applied field \(h\) which is taken to zero. The important feature is that in both cases the applied field is taken to zero after the approach to the thermodynamic limit.