In physics, the **Schrödinger picture** is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time.^{ [1] }^{ [2] } This differs from the Heisenberg picture 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 transformations and commutation relations between operators are preserved in the passage between the two pictures.

- Background
- The time evolution operator
- Definition
- Properties
- Differential equation for time evolution operator
- Summary comparison of evolution in all pictures
- See also
- Notes
- References

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 t_{0} to a state vector at time t, the time-evolution operator is commonly written , and one has

In the case where the 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 H; that is, .

In elementary quantum mechanics, the state of a quantum-mechanical system is represented by a complex-valued wavefunction *ψ*(*x*, *t*). 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 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 *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,

*Unitarity*

The time evolution operator must be unitary. This is the norm of the state ket must not change with time. That is,

Therefore,

*Identity*

When *t* = *t*_{0}, *U* is the identity operator, 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,

We drop the *t*_{0} index in the time evolution operator with the convention that *t*_{0} = 0 and write it as *U*(*t*). The Schrödinger equation is

where *H* is the Hamiltonian. Now using the time-evolution operator *U* to write ,

Since is a constant ket (the state ket at *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

If the Hamiltonian is independent of time, the solution to the above equation is^{ [note 1] }

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, after 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 Heisenberg picture.

For a time-independent Hamiltonian *H*_{S}, where *H*_{0,S} is the free Hamiltonian,

Evolution | Picture ( ) | ||

of: | Heisenberg | Interaction | Schrödinger |

Ket state | constant | ||

Observable | constant | ||

Density matrix | constant |

- ↑ At
*t*= 0,*U*(*t*) must reduce to the identity operator.

- ↑ Parker, C.B. (1994).
*McGraw Hill Encyclopaedia of Physics*(2nd ed.). McGraw Hill. pp. 786, 1261. ISBN 0-07-051400-3. - ↑ Y. Peleg; R. Pnini; E. Zaarur; E. Hecht (2010).
*Quantum mechanics*. Schuam's outline series (2nd ed.). McGraw Hill. p. 70. ISBN 978-0-07-162358-2.

In quantum mechanics, the **Hamiltonian** of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's *energy spectrum* or its set of *energy eigenvalues*, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

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 space which is a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces, and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.

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 the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

A **density matrix** is a matrix that describes the statistical state, whether pure or mixed, of a system in quantum mechanics. The probability for any outcome of any well-defined measurement upon a system can be calculated from the density matrix for that system. The extreme points in the set of density matrices are the pure states, which can also be written as state vectors or wavefunctions. Density matrices that are not pure states are *mixed states.* Any mixed state can be represented as a convex combination of pure states, and so density matrices are helpful for dealing with statistical ensembles of different possible preparations of a quantum system, or situations where a precise preparation is not known, as in quantum statistical mechanics.

In physics, an **operator** is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

In physics, the **Heisenberg picture** is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.

**Matrix mechanics** is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum jumps supplanted the Bohr model's electron orbits. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, as manifest in Dirac's bra–ket notation.

In quantum mechanics, the **canonical commutation relation** is the fundamental relation between canonical conjugate quantities. For example,

The **adiabatic theorem** is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows:

In quantum mechanics, the **interaction picture** is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables. The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. Most field-theoretical calculations use the interaction representation because they construct the solution to the many-body Schrödinger equation as the solution to the free-particle problem plus some unknown interaction parts.

In quantum mechanics, a **two-state system** is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In quantum mechanics, the **Hellmann–Feynman theorem** relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

In quantum mechanics, the **momentum operator** is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:

The **Ehrenfest theorem**, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators *x* and *p* to the expectation value of the force on a massive particle moving in a scalar potential ,

In quantum physics, **unitarity** is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics. A **unitarity bound** is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that time evolution preserves inner products in Hilbert space.

In quantum mechanics, given a particular Hamiltonian and an operator with corresponding eigenvalues and eigenvectors given by , then the numbers are said to be **good quantum numbers** if every eigenvector remains an eigenvector of *with the same eigenvalue* as time evolves.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

The **Gell-Mann and Low theorem** is a theorem in quantum field theory that allows one to relate the ground state of an interacting system to the ground state of the corresponding non-interacting theory. It was proved in 1951 by Murray Gell-Mann and Francis E. Low. The theorem is useful because, among other things, by relating the ground state of the interacting theory to its non-interacting ground state, it allows one to express Green's functions as expectation values of interaction picture fields in the non-interacting vacuum. While typically applied to the ground state, the Gell-Mann and Low theorem applies to any eigenstate of the Hamiltonian. Its proof relies on the concept of starting with a non-interacting Hamiltonian and adiabatically switching on the interactions.

In quantum mechanics, **dynamical pictures** are the multiple equivalent ways to mathematically formulate the dynamics of a quantum system.

Different subfields of physics have different programs for determining the state of a physical system.

- Cohen-Tannoudji, Claude; Bernard Diu; Frank Laloe (1977).
*Quantum Mechanics (Volume One)*. Paris: Wiley. pp. 312–314. ISBN 0-471-16433-X. - Albert Messiah, 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 ISBN 0-471-88702-1 - L.D. Landau, E.M. Lifshitz (1977).
*Quantum Mechanics: Non-Relativistic Theory*. Vol. 3 (3rd ed.). Pergamon Press. ISBN 978-0-08-020940-1. Online copy - R. Shankar (1994);
*Principles of Quantum Mechanics*, Plenum Press, ISBN 978-0-306-44790-7 . - J. J. Sakurai (1993);
*Modern Quantum Mechanics*(Revised Edition), ISBN 978-0-201-53929-5 .

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