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Equations that apply in one picture do not necessarily hold in the others, because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others. Not all textbooks and articles make explicit which picture each operator comes from, which can lead to confusion.
The Schrödinger equation is often presented using quantities varying as functions of position, but as a vector-operator equation it has a valid representation in any arbitrary complete basis of kets in Hilbert space. As mentioned above, "bases" that lie outside the physical Hilbert space are also employed for calculational purposes.
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 ] U ( t ) = e − i H t / ℏ . {\displaystyle U(t)=e^{-iHt/\hbar }.}
Position space (also real space or coordinate space) is the set of all position vectors r in Euclidean space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all momentum ...
The step divides space in two parts: x < 0 and x > 0. In any of these parts the potential is constant, meaning the particle is quasi-free, and the solution of the Schrödinger equation can be written as a superposition of left and right moving waves (see free particle)
For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a vector bundle over configuration space, and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space. [42] The field equations for the de Broglie–Bohm theory in the relativistic case with spin can also be given for ...