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In the mathematical formulation of quantum mechanics, each system is associated with a complex Hilbert space such that each instantaneous state of the system is described by a unit vector in that space. This state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. As the state of a system generally changes over time, the state vector is a function of time. The Schrödinger equation provides a quantitative description of the rate of change of the state vector.
Using Dirac's bra-ket notation, we denote that instantaneous state vector at time t by |ψ(t)⟩. The Schrödinger equation is:
where i is the unit imaginary number, is Planck's constantPlanck's constant denoted h is a physical constant that is used to describe the sizes of quanta. It plays a central role in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory. It has a value of approximat divided by 2π, and the HamiltonianThe Hamiltonian denoted H has two distinct but closely related meanings. In classical mechanics, it is a function that describes the state of a mechanical system in terms of position and momentum variables (i. symplectic variables), which is the basis for H(t) is a self-adjoint operatorOn a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. By the finite-dimensional acting on the state space. The Hamiltonian describes the total energyThis article is about the scientific concept. Energy use by humans is discussed in other articles''. Energy generally and qualitatively speaking, is the property (or the quantity of the property) of doing things or supplying power. The expressions energy of the system. As with the forceIn physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. The concept appeared first in the second law of motion of classical mechanics. It is usually expressed by the equation F m · a where F is the force, occurring in Newton's second law, its exact form is not provided by the Schrödinger equation, and must be independently determined based on the physical properties of the system.
For every time-independent Hamiltonian H, there exist a set of quantum states, known as energy eigenstates, satisfying the eigenvalue equation
Such a state possesses a definite total energy, whose value E is the eigenvalue of the state vector with the Hamiltonian. This eigenvalue equation is referred to as the time-independent Schrödinger equation. Self-adjoint operatorOn a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. By the finite-dimensionals such as the Hamiltonian have the property that their eigenvalues are always real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays, as we would expect since the energy is a physically observable quantity.
On inserting the time-independent Schrödinger equation into the full Schrödinger equation, we get
It is easy to solve this equation if we assume that H is not dependent in t. One finds that the state vectors of the energy eigenstates change by only a complex phaseThe phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). The phase may be measured as a time, distance,:
Energy eigenstates are convenient to work with because their time-dependence is so simple; that is why the time-independent Schrödinger equation is so useful. We can always choose a set of instantaneous energy eigenstates whose state vectors
(The last equation enforces the requirement that |ψ(t)>, like all state vectors, must be a unit vector.) Applying the Schrödinger equation to each side of the first equation, and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain
Therefore, if we know the decomposition of |ψ(t)> into the energy basis at time t = 0, its value at any subsequent time is given simply by
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