Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 109

Let us first assume that the wave function χ(r, t) changes smoothly from χl(r) to χ_l′(r) according to:

(63)

where the modulus of cl (t) decreases with time from 1 to 0, whereas cl′ (t) does the opposite. Normalization imposes that at all times t:

(64)

In such a state, let us show that the numerical density n(r, φ, z;t) now depends on φ (this was not the case for either states l or l′ separately). The transverse dependence of the density as a function of the variables r and z, is barely affected³. The variations of n(r, φ, z; t) are given by:

(65)

where c.c. stands for the complex conjugate of the preceding factor. The first two terms are independent of φ, and are just a weighted average of the densities associated with each of the states l and l′. The last term oscillates as a function of φ with an amplitude |cl (t)| × |cl′ (t)|, which is only zero if one of the two coefficients cl (t) or cl′ (t) is zero. Calling φl the phase of the coefficient cl (t) this last term is proportional to:

(66)

Whatever the phases of the two coefficients cl (t) and cl′ (t), the cosine will always oscillate between — 1 and 1 as a function of φ. Adjusting those phases, one can deliberately change the value of φ for which the density is maximum (or minimum), but this will always occur somewhere on the circle. Superposing two states necessarily modulates the density.

Let us evaluate the consequences of this density modulation on the internal repulsive interaction energy of the fluid. As we did in relation (15), we use for the interaction energy the zero range potential approximation, and insert it in expression (15) of Complement C_XV. Taking into account the normalization (17) of the wave function, we get:

(67)

We must now include the square of (65) in this expression, which will yield several terms. The first one, in |cl (t)|⁴, leads to the contribution:

(68)

where is the interaction energy for the state χl(r). The second contribution is the similar term for the state l′, and the third one, a cross term in 2|cl (t)|² |cl′ (t)|². Assuming, to keep things simple, that the densities associated with the states l and l′ are practically the same, the sum of these three terms is just:

(69)

Up to now, the superposition has had no effect on the repulsive internal interaction energy. As for the cross terms between the terms independent of φ in (65) and the terms in e^{±i(l – l′)φ}, they will cancel out when integrated over φ. We are then left with the cross terms in e^{±i(l – l′)φ} × e^{∓;i(l – l′)φ}, whose integral over φ yields:

(70)

Assuming as before that the densities associated with the states l and l′ are practically the same, we obtain, after integration over r and z:

(71)

Adding (69), we finally obtain:

(72)

We have shown that the density modulation associated with the superposition of states always increases the internal repulsion energy: this modulation does lower the energy in the low density region, but the increase in the high energy region outweighs the decrease (since the repulsive energy is a quadratic function of the density). The internal energy therefore varies between and the maximum (3/2) , reached when the moduli of cl (t) and cl′ (t) are both equal to .