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{{short description|Formulation of quantum mechanics}}
{{Quantum mechanics|cTopic=Formulations}}
In [[Physics:Physics|physics]], the '''Schrödinger picture''' or '''Schrödinger representation''' is a [[Physics:Dynamical pictures|formulation]] of [[Physics:Quantum mechanics|quantum mechanics]] 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 <math>V</math> changes).<ref>{{cite book|title=McGraw Hill Encyclopaedia of Physics|first1=C.B.|last1=Parker|edition=2nd|publisher=McGraw Hill|pages=[https://archive.org/details/mcgrawhillencycl1993park/page/786 786, 1261]|year=1994|isbn=0-07-051400-3|url-access=registration|url=https://archive.org/details/mcgrawhillencycl1993park/page/786}}</ref><ref>{{cite book |author1=Y. Peleg |author2=R. Pnini |author3=E. Zaarur |author4=E. Hecht |title=Quantum mechanics|edition=2nd|publisher=McGraw Hill|series=Schuam's outline series|year=2010|isbn=978-0-07-162358-2|page=70}}</ref> This differs from the [[Physics:Heisenberg picture|Heisenberg picture]] which keeps the states constant while the observables evolve in time, and from the [[Physics:Interaction picture|interaction picture]] in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as [[Physics:Active and passive transformation|active and passive transformation]]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 [[Physics:Unitarity|unitary operator]], the [[Time evolution|time evolution operator]]. For time evolution from a state vector <math>|\psi(t_0)\rangle</math> at time {{mvar|t}}<sub>0</sub> to a state vector <math>|\psi(t)\rangle</math> at time {{mvar|t}}, the time-evolution operator is commonly written <math>U(t, t_0)</math>, and one has
:<math>|\psi(t)\rangle = U(t, t_0) |\psi(t_0)\rangle.</math>
In the case where the [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]] {{mvar|H}} of the system does not vary with time, the time-evolution operator has the form
:<math> U(t, t_0) = e^{-iH\cdot (t-t_0)/\hbar},</math>
where the exponent is evaluated via its [[Taylor series]].

The Schrödinger picture is useful when dealing with a time-independent Hamiltonian {{mvar|H}}; that is, <math>\partial_t H=0 </math>.

==Background==

In elementary quantum mechanics, the [[Physics:Quantum state|state]] of a quantum-mechanical system is represented by a complex-valued wavefunction {{math|''ψ''(''x'', ''t'')}}. More abstractly, the state may be represented as a state vector, or [[Bra–ket notation|''ket'']], <math>| \psi \rangle</math>. This ket is an element of a ''[[Hilbert space]]'', a vector space containing all possible states of the system. A quantum-mechanical [[Physics:Operator#Operators in quantum mechanics|operator]] is a function which takes a ket  <math>| \psi \rangle</math> and returns some other ket  <math>| \psi' \rangle</math>.

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 [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]] may be in a state  <math>| \psi \rangle</math> for which the [[Physics:Expectation value (quantum mechanics)|expectation value]] of the momentum, <math>\langle \psi | \hat{p} | \psi \rangle</math>, oscillates sinusoidally in time. One can then ask whether this sinusoidal oscillation should be reflected in the state vector  <math>| \psi \rangle</math>, the momentum operator <math>\hat{p}</math>, 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''<sub>0</sub>) is defined as the operator which acts on the ket at time ''t''<sub>0</sub> to produce the ket at some other time ''t'':
<math display="block"> | \psi(t) \rangle = U(t,t_0) | \psi(t_0) \rangle.</math>

For bras, 
<math display="block"> \langle \psi(t) | = \langle \psi(t_0) |U^{\dagger}(t,t_0).</math>

===Properties===
;''Unitarity''
:The time evolution operator must be [[Unitary operator|unitary]]. For the [[Norm (mathematics)|norm]] of the state ket must not change with time. That is, <math display="block"> \langle \psi(t)| \psi(t) \rangle = \langle \psi(t_0)|U^{\dagger}(t,t_0)U(t,t_0)| \psi(t_0) \rangle  = \langle \psi(t_0) | \psi(t_0) \rangle.</math> Therefore, <math display="block"> U^{\dagger}(t,t_0)U(t,t_0)=I.</math>
;''Identity''
:When ''t''&nbsp;= ''t''<sub>0</sub>, ''U'' is the identity operator, since <math display="block"> | \psi(t_0) \rangle = U(t_0,t_0) | \psi(t_0) \rangle.</math>
;''Closure''
:Time evolution from ''t''<sub>0</sub> to ''t'' may be viewed as a two-step time evolution, first from ''t''<sub>0</sub> to an intermediate time ''t''<sub>1</sub>, and then from ''t''<sub>1</sub> to the final time ''t''. Therefore, <math display="block">U(t, t_0) = U(t, t_1) U(t_1, t_0).</math>

===Differential equation for time evolution operator===
We drop the ''t''<sub>0</sub> index in the time evolution operator with the convention that {{math|1=''t''<sub>0</sub> = 0}} and write it as ''U''(''t''). The [[Physics:Schrödinger equation|Schrödinger equation]] is
<math display="block"> i \hbar \frac{\partial}{\partial t} |\psi(t)\rangle = H |\psi(t)\rangle,</math>
where ''H'' is the [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]]. Now using the time-evolution operator ''U'' to write <math>|\psi(t)\rangle = U(t) |\psi(0)\rangle</math>,
<math display="block"> i \hbar {\partial \over \partial t} U(t) | \psi (0) \rangle = H U(t)| \psi (0)\rangle.</math>

Since <math>|\psi(0)\rangle</math> is a constant ket (the state ket at {{math|1=''t'' = 0}}), and since the above equation is true for any constant ket in the Hilbert space, the time evolution operator must obey the equation
<math display="block"> i \hbar \frac{\partial}{\partial t} U(t) = H U(t).</math>

If the Hamiltonian is independent of time, the solution to the above equation is<ref group="note">At {{nowrap|''t'' {{=}} 0}},  ''U''(''t'') must reduce to the identity operator.</ref>
<math display="block"> U(t) = e^{-iHt / \hbar}.</math>

Since ''H'' is an operator, this exponential expression is to be evaluated via its [[Taylor series]]:
<math display="block"> e^{-iHt / \hbar} = 1 - \frac{iHt}{\hbar} - \frac{1}{2} \left(\frac{Ht}{\hbar}\right)^2 + \cdots .</math>

Therefore,
<math display="block">| \psi(t) \rangle = e^{-iHt / \hbar} | \psi(0) \rangle.</math>

Note that <math>|\psi(0)\rangle</math> is an arbitrary ket. However, if the initial ket is an eigenstate of the Hamiltonian, with eigenvalue ''E'':
<math display="block">| \psi(t) \rangle = e^{-iEt / \hbar} | \psi(0) \rangle.</math>

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
<math display="block"> U(t) = \exp\left({-\frac{i}{\hbar} \int_0^t H(t')\, dt'}\right),</math>

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
<math display="block"> U(t) = \mathrm{T}\exp\left({-\frac{i}{\hbar} \int_0^t H(t')\, dt'}\right),</math>
where T is time-ordering operator, which is sometimes known as the [[Physics:Dyson series|Dyson series]], after [[Biography:Freeman Dyson|Freeman Dyson]].

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 [[Physics:Heisenberg picture|Heisenberg picture]].

==Summary comparison of evolution in all pictures==

For a time-independent Hamiltonian ''H''<sub>S</sub>, where ''H''<sub>0,S</sub> is the free Hamiltonian,

==See also==
*[[Physics:Hamilton–Jacobi equation|Hamilton–Jacobi equation]]
*[[Physics:Interaction picture|Interaction picture]]
*[[Physics:Heisenberg picture|Heisenberg picture]]
*Phase space formulation
*[[POVM]]
*[[Physics:Mathematical formulation of quantum mechanics|Mathematical formulation of quantum mechanics]]
*[[Physics:Schrödinger functional|Schrödinger functional]]

==Notes==
{{reflist|group="note"}}
{{reflist}}

==References==
* {{cite book
  | last = Cohen-Tannoudji
  | first = Claude
  |author2=Bernard Diu |author3=Frank Laloe
   | title = Quantum Mechanics (Volume One)
  | publisher = Wiley
  | year = 1977
  | location = Paris
  | pages = 312–314
  | isbn = 0-471-16433-X }}
* [[Biography:Albert Messiah|Albert Messiah]], 1966. ''Quantum Mechanics'' (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
*[[Biography:Eugen Merzbacher|Merzbacher E.]], ''Quantum Mechanics'' (3rd ed., John Wiley 1998) p.&nbsp;430–1 {{ISBN|0-471-88702-1}}
*{{cite book
 |author=[[Biography:Lev Landau|L.D. Landau]], [[Biography:Evgeny Lifshitz|E.M. Lifshitz]]
 |year=1977
 |title=Quantum Mechanics: Non-Relativistic Theory
 |edition=3rd |volume=3
 |publisher=Pergamon Press
 |isbn=978-0-08-020940-1
}} [https://archive.org/details/QuantumMechanics_104 Online copy] 
*[[Biography:Ramamurti Shankar|R. Shankar]] (1994); ''Principles of Quantum Mechanics'', Plenum Press, {{ISBN|978-0-306-44790-7}} .
*[[Biography:J. J. Sakurai|J. J. Sakurai]] (1993); ''[[Physics:Modern Quantum Mechanics|Modern Quantum Mechanics]]'' (Revised Edition),  {{ISBN|978-0-201-53929-5}} .

 
{{Quantum mechanics topics}}

{{DEFAULTSORT:Schrodinger Picture}}
[[Category:Foundational quantum physics]]

{{Sourceattribution|Schrödinger picture}}