Abstract
Quantum mechanics, as we know it so far, deals with invariant particle numbers,
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
M. Born, W. Heisenberg, P. Jordan, Z. Phys. 35, 557 (1926); P.A.M. Dirac, Proc. Roy. Soc. London A 114, 243 (1927); P. Jordan, E. Wigner, Z. Phys. 47, 631 (1928).
- 2.
Recall the canonical commutation relations \([x_{i}(t),p_{j}(t)] =\mathrm{ i}\hbar \delta _{ij}\), \([x_{i}(t),x_{j}(t)] = 0\), \([p_{i}(t),p_{j}(t)] = 0\) in the Heisenberg picture of quantum mechanics. It is customary to dismiss a factor of 2 in the (anti-)commutation relations (17.1), which otherwise would simply reappear in different places of the quantized Schrödinger theory.
- 3.
For another composite operator we can also define an integrated current density through \(\boldsymbol{I}_{q}(t) =\int \! d^{3}\boldsymbol{x}\,\boldsymbol{j}_{q}(\boldsymbol{x},t) = q\boldsymbol{P}(t)/m\), where the last equation follows from (16.24). However, recall that \(\boldsymbol{j}_{q}(\boldsymbol{x},t)\) is a current density, but it is not a current per volume, and therefore \(\boldsymbol{I}_{q}(t)\) is not an electric current but comes in units of e.g. Ampère meter. It is related to charge transport like momentum \(\boldsymbol{P}(t)\) is related to mass transport.
- 4.
For convenience, we have chosen the time when both pictures coincide as t 0 = 0.
- 5.
Recall that there are two time evolution operators in the Dirac picture. The free time evolution operator U 0(t − t′) evolves the operators \(\psi (\boldsymbol{x},t) = U_{0}^{+}(t - t')\psi (\boldsymbol{x},t')U_{0}(t - t')\), while U D (t, t′) evolves the states.
- 6.
- 7.
Formal substitution e.g. of two-particle tensor product states of definite spin for two identical particles,
$$\displaystyle{ \langle \boldsymbol{x},\sigma;\boldsymbol{ x}',\sigma '\vert \Phi _{n,n'}\rangle = \frac{\langle \boldsymbol{x}\vert \Phi _{n}\rangle \langle \boldsymbol{x}'\vert \Phi _{n'}\rangle \mp \delta _{\sigma \sigma '}\langle \boldsymbol{x}\vert \Phi _{n'}\rangle \langle \boldsymbol{x}'\vert \Phi _{n}\rangle } {\sqrt{2(1 +\delta _{nn' } \delta _{\sigma \sigma '})}} }$$(17.77)(cf. equation (17.66) for a = a′, \(\rho =\sigma\), \(\rho ' =\sigma '\)) and projection onto effective single particle equations using orthonormality of single particle wave functions yields exchange terms. However, the resulting equations are not identical with the Hartree-Fock equations from Problem 17.7, because E in equation (17.76) is the total energy of the system, whereas the Lagrange multipliers ε n in Hartree-Fock equations do not add up to the total energy of a many particle system, see Problem 17.7b. The formal nature of the substitution (17.77) is emphasized because we know that solutions of equation (17.76) do not factorize in single particle tensor products.
- 8.
As derived, this result applies to every 2-particle interaction potential. The most often studied case in atomic, molecular and condensed matter physics is the Coulomb interaction between electrons, and therefore the standard (non-exchange) interaction term is simply denoted as the Coulomb term.
- 9.
W. Heisenberg, Z. Phys. 38, 411 (1926); Z. Phys. 39, 499 (1926).
- 10.
P.A.M. Dirac, Proc. Roy. Soc. London A 123, 714 (1929).
- 11.
Heisenberg had introduced exchange integrals in 1926, and he published an investigation of ferromagnetism based on the exchange interaction (17.81) in 1928 (Z. Phys. 49, 619 (1928)). However, the effective Hamiltonian (17.84) was introduced by Dirac in the previously mentioned reference in 1929. Therefore a better name for (17.84) would be Dirac-Heisenberg Hamiltonian.
- 12.
- 13.
Bibliography
J. Callaway, Quantum Theory of the Solid State (Academic press, Boston, 1991)
P. Fulde, Electron Correlations in Molecules and Solids, 2nd edn. (Springer, Berlin, 1993)
M. Getzlaff, Fundamentals of Magnetism (Springer, Berlin, 2008)
H. Ibach, H. Lüth, Solid State Physics – An Introduction to Principles of Materials Science, 3rd edn. (Springer, Berlin, 2003)
C. Kittel, Quantum Theory of Solids, 2nd edn. (Wiley, New York, 1987)
O. Madelung, Introduction to Solid-State Theory (Springer, Berlin, 1978)
L. Marchildon, Quantum Mechanics: From Basic Principles to Numerical Methods and Applications (Springer, New York, 2002)
F. Schwabl, Quantum Mechanics, 4th edn. (Springer, Berlin, 2007)
F. Schwabl, Advanced Quantum Mechanics, 4th edn. (Springer, Berlin, 2008)
J. Stöhr, H.C. Siegmann, Magnetism – From Fundamentals to Nanoscale Dynamics (Springer, New York, 2006)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Dick, R. (2016). Non-relativistic Quantum Field Theory. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25675-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-25675-7_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25674-0
Online ISBN: 978-3-319-25675-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)