Abstract
Up to now we have only considered static situations (except for the reduction of the state vector when a measurement takes place). We now discuss the time dependence of the state vector, which requires a new principle. The resultant time-dependent Schrödinger equation is solved exactly for simple (spin) cases and in perturbation theory. The notion of transition probability yields physical meaning to non-diagonal matrix elements and allows us to present the energy–time uncertainty relation, together with the concept of mean lifetime (Sect. 9.5).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
As Jun John Sakurai points out: “Ironically, in the historical development of wave mechanics both L. de Broglie and E. Schrödinger were guided by a kind of covariant analogy between energy and time on the one hand and momentum and position (spatial coordinate) on the other. Yet when we now look at quantum mechanics in its finished form, there is no trace of a symmetrical treatment between time and space. The relativistic quantum theory of fields does treat the time and space coordinates on the same footing, but it does so at the expense of demoting position from the status of being an observable to that of being just a parameter.” [63], Chap. 2. Nevertheless, and although the commutation relation \([\hat{x},\hat{p}]\) has been postulated in the present text, it has also been derived in the non-relativistic limit of Lorentz transformations [18], thus suggesting a deeper link between relativity and quantum mechanics.
- 2.
The evolution is valid only for the Hamiltonian basis. Therefore, the expression \(\Psi (t) =\sum\limits_{i}{c}_{i}{\varphi }_{i}\text{ exp}(-\mathrm{i}{q}_{i}t/\hslash )\) makes no sense if \({\varphi }_{i}\), q i are not eigenstates and eigenvalues of the Hamiltonian.
- 3.
The density of states is given in (7.21) for the free particle case. A similar procedure is carried out for photons in (9.60).
- 4.
However, it has a different origin (see footnote 1).
- 5.
We ignore the ground state energy of the radiation field.
- 6.
These expressions have been derived using a similar procedure to the one used to obtain (7.21). A factor of 2, which was included in (7.21) due to spin, is not needed in (9.60). It reappears in (9.66), where the two directions of polarization are taken into account.
- 7.
\([\hat{\mathbf{p}},{\hat{\mathbf{A}}}_{t}] = 0\) because of (9.49).
- 8.
An order of magnitude of τ may be obtained by equating the rate of radiation of an oscillating classical dipole with the ratio between the emitted energy \(\hslash \omega \) and the mean lifetime
$$\frac{{\omega }^{4}{D}_{0}} {3{c}^{2}} = \frac{\hslash \omega } {\tau } \rightarrow \tau = \frac{3\hslash {c}^{2}} {{\omega }^{3}{D}_{0}}.$$The amplitude of the dipole oscillation is approximated by \({D}_{0} \approx -e{a}_{0}\). If the transition energy is assumed to be 10 eV, we obtain an estimated value \(\tau = O(1{0}^{-10}\) s).
- 9.
Laser, light amplification by stimulated emission of radiation; maser, microwave amplification by stimulated emission of radiation. These two devices differ in the range of electromagnetic frequency in which they operate.
References
B.C. Olschak: Buthan. Land of Hidden Treasures. Photography by U. and A. Gansser (Stein & Day, New York 1971).
M. Planck: Verh. Deutsch. Phys. Ges. 2, 207, 237 (1900).
A. Einstein: Ann. der Phys. 17, 132 (1905).
A.H. Compton: Phys. Rev. 21, 483 (1923).
C.L. Davisson and L.H. Germer: Nature 119, 528 (1927); G.P. Thomson: Proc. Roy. Soc. A 117, 600 (1928).
H. Geiger and E. Mardsen: Proc. Roy. Soc. A 82, 495 (1909); E. Rutherford: Phil. Mag. 21, 669 (1911).
J. Balmer: Verh. Naturf. Ges. Basel 7, 548, 750 (1885); Ann. der Phys. und Chem. 25, 80 (1885).
N. Bohr: Phil. Mag. 25, 10 (1913); 26, 1 (1913); Nature 92, 231 (1913).
W. Heisenberg: Zeitschr. Phys. 33, 879 (1925).
M. Born, W. Heisenberg and P. Jordan: Zeitschr. Phys. 35, 557 (1926).
P.A.M. Dirac: Proc. Roy. Soc. A 109, 642 (1925).
E. Schrödinger: Ann. der Phys. 79, 361, 489 (1926); 80, 437 (1926); 81, 109 (1926).
D.F. Styer et al.: Am J. Phys. 70, 288 (2002).
J. Schwinger: Quantum Mechanics. Symbolism of Atomic Measurements, ed. by B.G Englert (Springer-Verlag, Berlin, Heidelberg, New York 2001) Chap. 1.
P.A.M. Dirac: The Principles of Quantum Mechanics (Oxford University Press, Amen House, London 1930).
A. Einstein, B. Podolsky and N. Rosen: Phys. Rev. 47, 777 (1935).
O. Stern and W. Gerlach: Zeitschr. Phys. 8, 110 (1921); 9, 349 (1922).
G. Kaiser: J. Math. Phys. 22, 705 (1981).
D.F. Styer: Am. J. Phys. 64, 31 (1996).
J. Roederer: Information and its Role in Nature (Springer-Verlag, Berlin, Heidelberg, New York 2005).
N. Bohr: The Philosophical Writings of Niels Bohr (Ox Bow Press, Woodbridge, Connecticut 1987).
N.D. Mermin: Am. J. Phys. 71, 23 (2003).
R.F. Feynman, R.B. Leighton and M. Sands: The Feynman Lectures on Physics. Quantum Mechanics [Addison-Wesley, Reading (Massachusetts), London, New York, Dallas, Atlanta, Barrington (Illinois) 1965] Chap. 5.
A. Zeilinger, R. Gähler, C.G. Shull, W. Treimer and W. Hampe: Rev. Mod. Phys. 60, 1067 (1988).
O. Nairz, M. Arndt and A. Zeilinger: Am. J. Phys. 71, 319 (2003).
W. Heisenberg: Zeitschr. Phys. 43, 172 (1927).
B. Cougnet, J. Roederer and P. Waloshek: Z. für Naturforschung A 7, 201 (1952).
W. Wootters and W. Zurek: Nature 299, 802 (1982); Phys. Today, 76 (Feb. 2009).
R.S. Mulliken: Nature 114, 350 (1924).
E.T. Jaynes and F.W. Cummings: Proc. I.E.E.E. 51, 81 (1963).
M. Born: Zeitschr. Phys. 37, 863 (1926); 38, 499 (1926).
L. de Broglie: C. R. Acad. Sci. Paris 177, 507, 548 (1923).
G. Binning and H. Rohrer: Rev. Mod. Phys. 71, S324 (1999) and references contained therein.
F. Bloch: Zeitschr. Phys. 52, 555 (1928).
G.E. Uhlenbeck and S.A. Goudsmit: Nature 113, 953 (1925); 117, 264 (1926).
W. Pauli: Zeitschr. Phys. 43, 601 (1927).
S. Haroche and J.M. Raimond: Exploring the Quantum. Atoms, Cavities and Photons (Oxford University Press, Oxford (2010).
Å. Bohr and B. Mottelson: Nuclear Structure (W.A. Benjamin, New York, Amsterdam 1969)
R.P. Martinez–y–Romero, H.N. Nuñez–Yepez and A.L. Salas–Brito: Am. J. Phys. 75, 629 (2007).
A.K. Grant and J.L. Rosner: Am. J. Phys. 62, 310 (1994).
D.A. McQuarrie: Quantum Chemistry (University Science Books, Herndon, Virginia 1983) Fig. 6-12.
W. Pauli: Zeitschr. Phys. 31, 625 (1925).
K.L. Jones et al.: Nature 465, 454 (2010).
M. Kastner: Physics Today 46, 24 (1993).
R. Fitzgerald: Physics Today 57, 22 (2004) and references contained therein.
A. Einstein: Sitz. Ber. Preuss. Ak. Wiss. 2, 261 (1924); 3 (1925).
D. Kleppner: Physics Today 49, 11 (1996); F. Dalfovo, S. Giorgini, L.P. Pitaevckii and S. Stringari: Rev. Mod. Phys. 71, 463 (1999).
M.H. Anderson, J.R. Enscher, M.R. Matthews, C.E. Wieman and E.A. Cornell: Science 269, 198 (1995); J.R. Enscher, D.S. Jin, M.R. Matthews, C.E. Wieman and E.A. Cornell: Phys. Rev. Lett. 77, 4984 (1996).
K. von Klitzing, G. Dorda and M. Pepper: Phys. Rev. Lett. 45, 494 (1980).
D.C. Tsui, H.L. Störmer and A.C. Gossard: Phys. Rev. Lett. 48, 1559 (1982); Phys. Rev. B 25, 1405 (1982).
R.E. Prange: Introduction to The Quantum Hall Effect, ed. by R.E. Prange and S.M. Girvin (Springer-Verlag, New York, Berlin, Heidelberg 1987) Fig. 1.2.
B.L. Halperin: Scientific American 254, Vol. 4, 52 (1986).
J.D. Jackson: Classical Electrodynamics (John Wiley & Sons, New York, Chichester, Brisbane, Toronto 1975) p. 574.
R.B. Laughlin: Phys. Rev. Lett. 50, 1395 (1983).
S.N. Bose: Zeitschr. Phys. 26, 178 (1924).
P.A.M. Dirac: Proc. Roy. Soc. A 112, 661 (1926).
E. Fermi: Zeitschr. Phys. 36, 902 (1926).
R. Eisberg and R. Resnick: Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (John Wiley & Sons, New York (1975).
R.P. Feynman: Phys. Rev. 76, 769 (1949).
C. Itzykson and J.B. Zuber: Quantum Field Theory (McGraw-Hill, New York, St. Louis, San Francisco, London 1980).
B.H. Brandow: Rev. Mod. Phys. 39, 771 (1967); E.M. Krenciglowa and T.T.S. Kuo: Nucl. Phys. A 240, 195 (1975).
C. Bloch and J. Horowitz: Nucl. Phys. 8, 91 (1958).
J.J. Sakurai: Modern Quantum Mechanics. Addison–Wesley Pub. Co., Reading, Massachusetts (1994).
A. Einstein: Verh. Deutsch. Phys. Ges. 18, 318 (1916); Mitt. Phys. Ges. Zürich 16, 47 (1916); Phys. Zeitschr. 18, 121 (1917).
T.H. Maiman: Nature 187, 493 (1960).
J. Bardeen, L.N. Cooper and J.R. Schrieffer: Phys. Rev. 106, 162 (1957).
C. Becchi, A. Rouet and R. Stora: Phys. Lett. B 52, 344 (1974).
M. Henneaux and C. Teitelboim: Quantization of Gauge Systems (Princeton University Press, Princeton, New Jersey 1992).
D.R. Bes and J. Kurchan: The Treatment of Collective Coordinates in Many-Body Systems. An Application of the BRST Invariance (World Scientific Lecture Notes in Physics, Vol. 34, Singapore, New Jersey, London, Hong Kong 1990); D.R. Bes and O. Civitarese: Am. J. Phys. 70, 548 (2002).
H. Kammerlingh Onnes, Comm. Phys. Lab. Univ. Leiden, Nos. 122 and 124 (1911).
B.D. Josephson, Phys. Lett. 1, 251 (1962).
R. Feynman and A. Hibbs: Quantum Mechanics and Path Integrals (Mc Graw–Hill Book Co., New York 1965).
P. Cartier and C. DeWitt–Morette: Functional Integration (Cambridge U. Press, New York 2007).
S. Carlip: Phys. Today 61, 61 (2008).
L.S. Schulman: Techniques and Applications of Path Integrals (John Wiley & Sons, New York 1981).
E. Schrödinger: Naturwissenschaften 23, 807, 823, 845 (1935)
A. Aspect: Introduction: John Bell and the Second Quantum Revolution, in J.S. Bell, “Speakable and Unspeakable in Quantum Mechanics”, Cambridge University Press (2008).
A. Zeilinger: Rev. Mod. Phys. 71, S288 (1999) and references contained therein
A. Zeilinger, G. Weihs, T. Jennewein and M. Aspelmeyer: Nature 433, 230 (2005).
D. Bohm: Quantum Theory. Prentice Hall, New Jersey (1951); Phys. Rev. 85, 166, 180 (1952).
J.S. Bell: Physics 1, 195 (1964).
N.D. Mermin: Physics Today, 38 (1985).
A. Aspect, P. Grangier and G. Roger: Phys. Rev. Lett. 47, 460 (1981); 49, 91 (1982).
G. Weihs, T. Jennewein, C. Simon, H. Weinfurter and A. Zeilinger: Phys. Rev. Lett. 81, 5039 (1998).
W. Nagourney, J. Sandberg and H. Dehmet, Phys. Rev. Lett. 56, 2797 (1986).
M. Brune et al.: Phys. Rev. Lett. 76, 1800 (1996).
M.A. Rowe et al.: Nature 409, 791 (2001).
A.D. O’Connell et al.: Nature 464, 697 (2010).
C.H. Bennet and G. Brassard: Proc. I.E.E.E. Int. Conf. on Computers, Systems and Signal Processing (IEEE Press, Los Alamos, California 1984) p. 175.
C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W. Wooters: Phys. Rev. Lett. 70, 1895 (1993).
D. Bouwmeester, J.W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger: Nature 390, 575 (1997).
E. Gerjuoy: Am. J. Phys. 73, 521 (2005).
P.W. Schor: Proc. 34th Annual Symp. Found. Comp. Scien. (FOCS), ed. by S. Goldwasser (IEEE Press, Los Alamitos, California 1994) p. 124.
I.L. Chuang, L.M.K. Vandersypen, X.L. Zhou, D.W. Leung and S. Lloyd: Nature 393, 143 (1998).
P.A.M. Dirac: Hungarian Ac. of Sc. Rep. KFK-62 (1977).
E. Schrödinger: Naturwissenschaftern. 23, 807, 823, 844 (1935).
R. Penrose: The Road to Reality (Alfred A. Knopf, New York, 2005, Chapter 29).
M. Schlosshauer: Decoherence and the Quantum–to–Classical Transition (Springer–Verlag, Berlin, Heidelberg 2007).
W.H. Zurek: Revs. Mod. Phys. 75, 715 (2003) and references contained therein; Phys. Today 49, 36 (1991).
J.P. Paz and W.H. Zurek: Environment-Induced Decoherence and the Transition from Classical to Quantum. In: Coherent Atomic Matter Waves, Les Houches Session LXXXII, ed. by R. Kaiser, C. Westbrook and F. Davids (Springer-Verlag, Berlin, Heidelberg, New York 2001) p. 533.
J. von Neumann: Matematische Grundlagen der Quantenmechanik (Springer, Berlin 1932).
L. Hackermüller, K. Hornberger, B. Brezger, A. Zeilinger, M. Arndt: Appl. Phys. B 77, 781 (2003)
E. Hobsbawn: The Age of Extremes. A History of the World, 1914–1991 (Vintage Books, New York 1996) p. 7.
O. Spengler: Der Untergang des Abendlandes. Umrisse einer Morphologie der Weltgeschichte, Vol. 1: Gestalt und Wirklichkeit (Munich, 1918). First English translation: The Decline of the West, Vol. 1: Form and Actuality (Knof, New York 1926).
H. Kragh: Quantum Generations. A History of Physics in the Twentieth Century (Princeton University Press, Princeton, New Jersey 1999).
A. Pais: ‘Subtle is the Lord...’ The Science and the Life of Albert Einstein (Oxford University Press, Oxford, New York, Toronto 1982). Inward Bound (Oxford University Press, Oxford, New York, Toronto 1986); Niels Bohr’s Times. In Physics, Philosophy and Politics (Clarendon Press, Oxford 1991).
P. Robertson: The Early Years. The Niels Bohr Institute 1921–1930 (Akademisk Forlag. Universitetsforlaget i København, Denmark 1979).
G. Kirchhoff: Ann. Phys. Chem. 109, 275 (1860).
W. Wien: Sitz. Ber. Preuss. Ak. Wiss. 55 (1893); Ann. Physik 58, 662 (1896).
G. Kirchhoff and R. Bunsen: Ann. Phys. Chem. 110, 160 (1860).
R. Millikan: Phys. Rev. 4, 73 (1914); 6, 55 (1915).
E. Lawrence and J. Beams: Phys. Rev. 32, 478 (1928); A. Forrester, R. Gudmundsen and P. Johnson: Phys. Rev. 90, 1691 (1955).
J. Clauser: Phys. Rev. D 9, 853 (1974).
H.G.J. Moseley: Nature 92, 554 (1913); Phil. Mag. 26, 1024 (1913); 27, 703 (1914).
J. Frank and H. Hertz: Verh. Deutsch. Phys. Ges. 16, 457 (1914).
A. Sommerfeld: Sitz. Ber. Bayer. Akad. Wiss. 459 (1915).
L.H. Thomas: Nature 117, 514 (1926); Phil. Mag. 3, 1 (1927).
M. Born and P. Jordan: Zeitschr. Phys. 34, 858 (1925).
W. Heisenberg: Zeitschr. Phys. 38, 499 (1926).
P.A.M. Dirac: Proc. Roy. Soc. A 117, 610 (1928); A 118, 351 (1928).
N. Bohr, H.A. Kramers and J.C. Slater: Phil. Mag. [6] 47, 785 (1924); Zeitschr. Phys. 24, 69 (1924).
N. Bohr: Nature 136, 65 (1935); Phys. Rev. 48, 696 (1935).
A. Einstein: Reply to Criticisms. In: Albert Einstein. Philosopher-Scientist, ed. by P.A Schilpp (Open Court Publishing, Peru, Illinois 2000) p. 663.
N. Bohr: Discussions with Einstein on Epistemological Problems in Atomic Physics. In Albert Einstein. Philosopher-Scientist, ed. by P.A Schilpp (Open Court Publishing, Peru, Illinois 2000) p. 199.
M. Tegmark and J.A. Wheeler: Scientific American 284, Vol. 2, 54 (2001).
J. Bell, Rev. Mod. Phys. 38, 447 (1966).
H.D. Zeh: Found. Phys. 1, 69 (1970).
W.H. Zurek: Phys. Rev. D 24, 1516 (1981); 26, 1862 (1982).
W.H. Zurek: Reduction of the Wavepacket: How Long does it take?. In: Frontiers in Nonequilibrium Statistical Mechanics, G.T. Moore and M.O. Scully, eds. (Plenum Press, New York, 1986) p.145
E. Joos and H.D. Zeh: Z. Phys. B: Condens. Matter 59, 223 (1985).
The NIST Reference on Constants, Units, and Uncertainty: Appendix 3. http:/physics.nist.gov/cuu/Units/units.html.
R.W. Richardson: Phys. Lett. 3, 277 (1963); J. Dukelsky, H.S. Lerma, L. Robledo, R. Rodriguez-Guzman and S.M. Rombouts: Phys. Rev. C 84, 061301 (2011).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bes, D.R. (2012). Time Dependence in Quantum Mechanics. In: Quantum Mechanics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20556-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-20556-9_9
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20555-2
Online ISBN: 978-3-642-20556-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)