The average values of the powers of |q→|n¯≡qn and |v→|n¯≡vn are commonly used physical quantities, where q→ and v→ are the three-dimensional momentum and velocity of a quark inside a meson, respectively. For example, the average kinetic energy μπ2=q2 is an important quantity in the heavy quark effective theory, and the expansion of vn is commonly used in the nonrelativistic quantum chromodynamics. In this paper, based on the instantaneous Bethe-Salpeter equation method, we calculate the average values qn and vn (n=1, 2, 3, 4) of a heavy quark inside S wave and P wave heavy-light mesons. We obtain the kinetic energy μπ2=0.455 GeV2 for the B meson, which is consistent with the experimental result 0.464±0.076 GeV2. For the Bs, D, and Ds mesons, the μπ2 are 0.530, 0.317, and 0.369 GeV2, respectively, and v2=0.0185, 0.0215, 0.121, and 0.140 for B, Bs, D, and Ds, respectively. We obtain some relationships, for example, q0−n(mS)≈q1−n(mS), q0+n(mP)≈q1+′n(mP′)>q1+n(mP)≈q2+n(mP), and qn(mS)<qn(mP) (m=1, 2, 3), etc. Published by the American Physical Society 2024
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