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r = tanh - - :::::: - -

(11.124)

X::::::-

(11.125)

How would interparticle interaction affect the magnetic properties of an electron gas Qualitatively speaking, the effect of a repulsive interaction will enhance the paramagnetism, because two electrons would prefer to be in a spatially antisymmetric wave function to minimize the repulsive energy. But an antisymmetric spatial wave function requires a symmetric spin wave function, which is a triplet state. Thus the repulsion tends to align the spins of the electrons. We demonstrate this effect by considering an imperfect gas of spin- ~ fermions at very low temperatures, so that the repulsive interaction can be characterized by a single parameter, the scattering length a, or effective hardsphere diameter. To first order in a, the energy for an N-particle system is given by (AA1) in the Appendix. We take our model to be defined by (11.102) plus the interaction energy:

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The condition for the validity of this formula is that k FI a I 1 where k F is the wave number of a particle at the Fermi level: k} = (3'7T 2n)2/3. Thus, the condition for validity is low density, i.e., na 3 1. The partition function is

4'7Tan N+N_]\

FIGURE 7.29 A security can continue to decline, even when its trendline changes from resistance to support.

(11.127) The notation is the same as that of (11.106). Proceeding in the same way as in the evaluation of (11.106), we obtain

NlogQN=pg(N+)

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(11.128)

max [g(N J]

(11.129)

- N[A(N+) +A(N-N+)]

(11.130)

It must be noted that (11.130) locates the point at which the curve g(N +) passes through a maximum. It is conceivable (and in fact true) that N+ may occur not ~ a maxim~ of the curve g(N +) but at the boundary of the range of N +' i.e., at N + = 0 or N + = N. We keep this in mind as we proceed. With v(N) defined as in (11.114), we rewrite (11.130) as

fund, because it is a double-inverse index fund. What that means is that if a particular market (in this case, the NASDAQ 100) goes down 1 percent, the value of the fund goes twice as far the opposite direction (up 2 percent). So in a bear market, a fund like this provides an opportunity for people to still increase their portfolio s value. This graph shows the fund during a relatively bullish phase of the stock market, when the fund s value declined from $97.50 to below $15. Notice how on three separate occasions the prices just barely kissed the trendline, only to be repelled each time. After the third kiss, prices got enough strength to push through to the other side, allowing the fund to form a basing pattern and stabilize at about $15. Now let s examine the opposite effect a stock in an uptrend that breaks beneath it.

(15.96)

(15.97)

The three equations (15.96) then become the same one:

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(15.98)

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