Schrödinger picture

Schrödinger picture
In physics, the Schrödinger picture or Schrödinger representation is a formulation of 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 changes). 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.
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 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 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 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, t0) is defined as the operator which acts on the ket at time t0 to produce the ket at some other time t:
For bras,
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 = t0, U is the identity operator, since
Closure
Time evolution from t0 to t may be viewed as a two-step time evolution, first from t0 to an intermediate time t1, and then from t1 to the final time t. Therefore,
Differential equation for time evolution operator
We drop the t0 index in the time evolution operator with the convention that 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 ), 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
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.
Summary comparison of evolution in all pictures
For a time-independent Hamiltonian HS, where H0,S is the free Hamiltonian,
See also
Hamilton–Jacobi equation
Interaction picture
Heisenberg picture
Phase space formulation
POVM
Mathematical formulation of quantum mechanics
Schrödinger functional
Notes
References
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
Online copy
R. Shankar (1994); Principles of Quantum Mechanics, Plenum Press, .
J. J. Sakurai (1993); Modern Quantum Mechanics (Revised Edition), .
Foundational quantum physics
Picture