In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the Schrödinger picture is a
formulation of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
in which the
state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may change if the potential
changes).
This differs from the
Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but ...
which keeps the states constant while the observables evolve in time, and from the
interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as
active and passive transformation
Active may refer to:
Music
* ''Active'' (album), a 1992 album by Casiopea
* Active Records, a record label
Ships
* ''Active'' (ship), several commercial ships by that name
* HMS ''Active'', the name of various ships of the British Royal ...
s and
commutation relations between operators are preserved in the passage between the two pictures.
In the
Schrödinger picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a
unitary operator, the
time evolution operator. For time evolution from a state vector
at time
0 to a state vector
at time , the time-evolution operator is commonly written
, and one has
:
In the case where the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
of the system does not vary with time, the time-evolution operator has the form
:
where the exponent is evaluated via its
Taylor series.
The Schrödinger picture is useful when dealing with a time-independent Hamiltonian ; that is,
.
Background
In elementary quantum mechanics, the
state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...
of a quantum-mechanical system is represented by a complex-valued
wavefunction . More abstractly, the state may be represented as a state vector, or
''ket'',
. This ket is an element of a ''
Hilbert space'', a vector space containing all possible states of the system. A quantum-mechanical
operator is a function which takes a ket
and returns some other ket
.
The differences between the Schrödinger and Heisenberg pictures of quantum mechanics revolve around how to deal with systems that evolve in time: the time-dependent nature of the system ''must'' be carried by some combination of the state vectors and the operators. For example, a
quantum harmonic oscillator may be in a state
for which the
expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of the momentum,
, oscillates sinusoidally in time. One can then ask whether this sinusoidal oscillation should be reflected in the state vector
, the momentum operator
, or both. All three of these choices are valid; the first gives the Schrödinger picture, the second the Heisenberg picture, and the third the interaction picture.
The time evolution operator
Definition
The time-evolution operator ''U''(''t'', ''t''
0) is defined as the operator which acts on the ket at time ''t''
0 to produce the ket at some other time ''t'':
For
bras
A broadband remote access server (BRAS, B-RAS or BBRAS) routes traffic to and from broadband remote access devices such as digital subscriber line access multiplexers (DSLAM) on an Internet service provider's (ISP) network. BRAS can also be refe ...
,
Properties
;''Unitarity''
:The time evolution operator must be
unitary. For the
norm of the state ket must not change with time. That is,
Therefore,
;''Identity''
:When ''t'' = ''t''
0, ''U'' is the
identity operator
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film) ...
, since
;''Closure''
:Time evolution from ''t''
0 to ''t'' may be viewed as a two-step time evolution, first from ''t''
0 to an intermediate time ''t''
1, and then from ''t''
1 to the final time ''t''. Therefore,
Differential equation for time evolution operator
We drop the ''t''
0 index in the time evolution operator with the convention that and write it as ''U''(''t''). The
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
is
where ''H'' is the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
. Now using the time-evolution operator ''U'' to write
,
Since
is a constant ket (the state ket at ), and since the above equation is true for any constant ket in the Hilbert space, the time evolution operator must obey the equation
If the Hamiltonian is independent of time, the solution to the above equation is
[At , ''U''(''t'') must reduce to the identity operator.]
Since ''H'' is an operator, this exponential expression is to be evaluated via its
Taylor series:
Therefore,
Note that
is an arbitrary ket. However, if the initial ket is an
eigenstate of the Hamiltonian, with eigenvalue ''E'':
The eigenstates of the Hamiltonian are ''stationary states'': they only pick up an overall phase factor as they evolve with time.
If the Hamiltonian is dependent on time, but the Hamiltonians at different times commute, then the time evolution operator can be written as
If the Hamiltonian is dependent on time, but the Hamiltonians at different times do not commute, then the time evolution operator can be written as
where T is
time-ordering operator, which is sometimes known as the
Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagra ...
, after
Freeman Dyson
Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...
.
The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. This is the
Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but ...
.
Summary comparison of evolution in all pictures
For a time-independent Hamiltonian ''H''
S, where ''H''
0,S is the free Hamiltonian,
See also
*
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
*
Interaction picture
*
Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but ...
*
Phase space formulation
The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...
*
POVM
In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) a ...
*
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which ...
Notes
References
*
*
Albert Messiah
Albert Messiah (23 September 1921, Nice – 17 April 2013, Paris) was a French physicist.
He studied at the Ecole Polytechnique.
He spent the Second World War in the Free France forces: he embarked on 22 June 1940 at Saint-Jean-de-Luz for Engla ...
, 1966. ''Quantum Mechanics'' (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
*
Merzbacher E., ''Quantum Mechanics'' (3rd ed., John Wiley 1998) p. 430–1
Online copy
*
R. Shankar
Ramamurti Shankar (born April 28, 1947) is the Josiah Willard Gibbs professor of Physics at Yale University, in New Haven, Connecticut.
Education
He received his B. Tech in electrical engineering from the Indian Institute of Technology Madras, I ...
(1994); ''Principles of Quantum Mechanics'', Plenum Press, .
*
J. J. Sakurai
was a Japanese-American particle physicist and theorist.
While a graduate student at Cornell University, Sakurai independently discovered the V-A theory of weak interactions.
He authored the popular graduate text ''Modern Quantum Mechanics'' ( ...
(1993); ''
Modern Quantum Mechanics'' (Revised Edition), .
{{DEFAULTSORT:Schrodinger Picture
Foundational quantum physics
Picture