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+ lf3mw 2 - vf --log (1 - e- PhWk ) (13.49)

(27T)3

~(w)

(13.50)

where tj = exp(27Tf3wjL), n j = (L/27T)Ij, and the contours of integration are circles about tj = 0. We can also write Q(P) The result is

(13.52)

(13.51)

When you choose Apply Studies, a dialog box like the one in Figure 6.4 appears. This is the technical studies editor, and it has four main components:

where Qo is the partition function for Vs = 0. Thus, (n k >is still given by (13.19). Taking the ensemble average of (13.44) and noting that Lkn k = 0, we obtain

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Anderson's Equations Consider pure superfluid flow in a hydrodynamic regime, in which the phase cP, and therefore the superfluid velocity vs ' varies slowly on an atomic scale. We shall perform coarse graining, by imagining the superfluid (i.e., liquid He 4 in its ground state), being divided into cells containing a large number of atoms, but which are small on a macroscopic scale. We assume that both the phase cp and the particle number N vary only slightly from cell to cell, and average their values over a cell. These average values will be used as macroscopic thermodynamic variables. First let us construct a state of definite phase cp by superposing states of definite particle number N. As a model we choose

Double-click the time slot for the appointment that you want to set. Outlook opens the Appointment window, at the Appointment tab.

ICP) =

iN of>lN)

(13.53)

--";=2=7T=~=N=--00

(CPIl/JICP)

+ 1)

Studies and My Study Sets. The leftmost one, Studies, is selected by default. This gives you access to the same list of technical studies as is available in the Technical Studies module. You can add a study in any

(13.54)

It is assumed that the phase of IN> has been so chosen such that the matrix elements of l/J are real. The important feature to note is that states of different N have definite relative phases (instead of random relative phases as in an ensemble). The states labeled by different cP are not independent, however, because they are not orthogonal to each other:

(CPlcp/)

If you did not select the correct time slot in Step 3, you can click the Start time and click a start time.

(13.55)

= cP - cp'. But states satisfying*

(13.56) are essentially orthogonal, and thus may be considered independent. Accordingly, in each of our coarse-grained cells, the values of cP and N are to be averaged over respective intervals ~cP and tiN, which are small on a macroscopic scale, and at the same time satisfy (13.56). The variables cP and N are then semiclassical coordinates for the coarse-grained cell, which we call the "superfluid coordinates." Their values are defined only to accuracies within the "phasespace" cells shown in Fig. 13.6. When the superfluid coordinates are uniform throughout the system, the superfluid is in equilibrium. Any variations in the coordinates induce a flow of

'This is an uncertainty relation similar to that between energy and time (t1 E t1 t ::> n), there being no operator whose eigenvalue is cp.

Fig. 13.6 Number of particles Nand superflUI': phase </> obey uncertainty relation Ii N Ii</> ~ 1. Sho\\!: here are minimal cells in "phase space."

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