Schrödinger picture
Encyclopedia
In physics
, the Schrödinger picture is a formulation of quantum mechanics
in which the state vector
s evolve in time, but the operators (observables and others) are constant. This differs from the Heisenberg picture
which keeps the states constant while the observables evolve in time. The two models are related as active and passive transformation
s and have the same measurement statistics.
A state function is a linear combination, or a superposition, of eigenstates. 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 called the time evolution operator
.
That is, this operator when acting on the state ket
at t0 gives the state ket at a later time t. For bras
, we have:
. This is because we demand that the norm
of the state ket must not change with time. That is,
Therefore,
can be written as:
Here H is the Hamiltonian
for the system. As
is a constant ket (the state ket at ), we see that the time evolution operator obeys the Schrödinger equation
: i.e.
If the Hamiltonian is independent of time, the solution to the above equation is:
Where we have also used the fact that at , U(t) must reduce to the identity operator. Therefore we get:
Note that
is an arbitrary ket. However, if the initial ket is an eigenstate of the Hamiltonian, with eigenvalue E, we get:
Thus we see that 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:
where T is time-ordering operator
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
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the Schrödinger picture is a formulation of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
in which the state vector
State vector
*A state vector in general control systems describes the observed states of an object in state space, e.g. in variables of the degrees of freedom for motion *A state vector in general control systems describes the observed states of an object in state space, e.g. in variables of the degrees of...
s evolve in time, but the operators (observables and others) are constant. This differs from the Heisenberg picture
Heisenberg picture
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. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...
which keeps the states constant while the observables evolve in time. The two models are related as active and passive transformation
Active and passive transformation
In the physical sciences, an active transformation is one which actually changes the physical position of a system, and makes sense even in the absence of a coordinate system whereas a passive transformation is a change in the coordinate description of the physical system . The distinction between...
s and have the same measurement statistics.
A state function is a linear combination, or a superposition, of eigenstates. In the Schrödinger picture, the state of a system evolves with time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
. The evolution for a closed quantum system is brought about by a unitary operator called the time evolution operator
Time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...
.
Definition
The time evolution operator U(t,t0) is defined as:That is, this operator when acting on the state ket
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
at t0 gives the state ket at a later time t. For bras
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
, we have:
Property 1
The time evolution operator must be unitaryUnitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
. This is because we demand that the norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
of the state ket must not change with time. That is,
Therefore,
Property 2
Clearly , the Identity operator. As:Property 3
Also time evolution from t0 to t may be viewed as time evolution from t0 to an intermediate time t1 and 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 equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
can be written as:
Here H is the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
for the system. As
is a constant ket (the state ket at ), we see that the time evolution operator obeys the Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
: i.e.
If the Hamiltonian is independent of time, the solution to the above equation is:
Where we have also used the fact that at , U(t) must reduce to the identity operator. Therefore we get:
Note that
is an arbitrary ket. However, if the initial ket is an eigenstate of the Hamiltonian, with eigenvalue E, we get:Thus we see that 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:
where T is time-ordering operator
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
Heisenberg picture
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. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...
.
Further reading
- Principles of Quantum Mechanics by R. Shankar, Plenum Press.