Momentum operator

In quantum mechanics, the momentum operator is the operator associated with the measurement of linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one dimension, the definition is: In quantum mechanics, the momentum operator is the operator associated with the measurement of linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one dimension, the definition is: where ħ is Planck's reduced constant, i the imaginary unit, and partial derivatives (denoted by ∂ {displaystyle partial } ) are used instead of a total derivative (d/dx) since the wave function is also a function of time. The 'hat' indicates an operator. The 'application' of the operator on a differentiable wave function is as follows: In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication by p, i.e. it is a multiplication operator, just as the position operator is a multiplication operator in the position representation. At the time quantum mechanics was developed in the 1920s, the momentum operator was found by many theoretical physicists, including Niels Bohr, Arnold Sommerfeld, Erwin Schrödinger, and Eugene Wigner. Its existence and form is sometimes taken as one of the foundational postulates of quantum mechanics. The momentum and energy operators can be constructed in the following way. Starting in one dimension, using the plane wave solution to Schrödinger's equation of a single free particle, where p is interpreted as momentum in the x-direction and E is the particle energy. The first order partial derivative with respect to space is