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In quantum mechanics, position and momentum are conjugate variables. For a single particle described in the position basis the momentum operator can be written as = =, where ∇ is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit. This is a commonly encountered form of the momentum operator, though the ...
The derivation in three dimensions is the same, except the gradient operator del is used instead of one partial derivative. In three dimensions, the plane wave solution to Schrödinger's equation is: = and the gradient is = + + = (+ +) = where e x, e y, and e z are the unit vectors for the three spatial dimensions, hence ^ =
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Symbol Name Meaning SI unit of measure nabla dot the divergence operator often pronounced "del dot" per meter (m −1) nabla cross the curl operator often pronounced "del cross" per meter (m −1) nabla: delta (differential operator)
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
In mechanics, a variable-mass system is a collection of matter whose mass varies with time. It can be confusing to try to apply Newton's second law of motion directly to such a system. [ 1 ] [ 2 ] Instead, the time dependence of the mass m can be calculated by rearranging Newton's second law and adding a term to account for the momentum carried ...
A plot of position and momentum variables as a function of time is sometimes called a phase plot or a phase diagram. However the latter expression, " phase diagram ", is more usually reserved in the physical sciences for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, which consists of ...
Left: intrinsic "spin" angular momentum S is really orbital angular momentum of the object at every point, right: extrinsic orbital angular momentum L about an axis, top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω) [6] bottom: momentum p and its radial position r from the axis.