Diploma thesis - Fachbereich Physik

3.3. GROUND-STATE ENERGY WITH EXTERNAL CURRENT 45
k
1
2
3
ǫ k
Aj(2j 2 + 3ω 3 )
2ω 6
2Bω 2 (4j 4 + 12j 2 ω 3 + 3ω 6 ) − A 2 (36j 4 + 54j 2 ω 3 + 11ω 6 )
8ω 10
Aj[3A 2 (36j 4 + 63j 2 ω 3 + 22ω 6 ) − 2Bω 2 (24j 4 + 66j 2 ω 3 + 31ω 6 )]
4ω 14
4 [36A 2 Bω 2 (112j 6 + 324j 4 ω 3 + 212j 2 ω 6 + 19ω 9 )
−4B 2 ω 4 (64j 6 + 264j 4 ω 3 + 248j 2 ω 6 + 21ω 9 )
−3A 4 (2016j 6 + 4158j 4 ω 3 + 2112j 2 ω 6 + 155ω 9 )]/(32ω 10 )
5 Aj[27A 4 (1728j 6 + 4158j 4 ω 3 + 2816j 2 ω 6 + 465ω 9 )
+4B 2 ω 4 (1536j 6 + 6408j 4 ω 3 + 7072j 2 ω 6 + 1683ω 9 )
−12A 2 Bω 2 (3456j 6 + 10908j 4 ω 3 + 9176j 2 ω 6 + 1817ω 9 )]/(32ω 22 )
6 [8B 3 ω 6 (1536j 8 + 8544j 6 ω 3 + 14144j 4 ω 6 + 6732j 2 ω 9 + 333ω 12 )
−4A 2 B 2 ω 4 (103680j 8 + 454032j 6 ω 3 + 584928j 4 ω 6 + 221706j 2 ω 9 + 11827ω 12 )
+6A 4 Bω 2 (285120j 8 + 991224j 6 ω 3 + 1024224j 4 ω 6 + 323544j 2 ω 9 + 15169ω 12 )
−A 6 (1539648j 8 + 4266108j 6 ω 3 + 3649536j 4 ω 6 + 979290j 2 ω 9 + 39709ω 12 )]/(128ω 26 )
Table 3.2: Energy corrections for the ground-state energy of the oscillator with cubic and
quartic anharmonicity in the presence of an external current up to the 6th order.
Thus, the effective potential is obtained by solving the differential equation
V eff (X) = E (V eff ′ (X)) + V eff ′ (X)X . (3.41)
To this end, the effective potential is expanded in the coupling constant,
V eff (X) =
∞∑
g k V k (X) , (3.42)
k=0
and each order V k (X) is assumed to be a polynomial in the background X:
V k (X) =
∑k+2
m=0
C (k)
m X m . (3.43)
Using the result for the energy (3.31), where the first orders of ǫ k are given by Tab. 3.2, and
inserting the ansatz (3.42), (3.43) into the differential equation (3.41) permits us to obtain
the effective potential by performing a coefficients comparison, first in the relevant order of
the coupling constant g, and then for each order of X. It turns out that for k being even or