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Published byElwin Booker Modified about 1 year ago

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Operators - are performed on functions -are performed on vector functions and have directional qualities as well. These are referred to as vector operators. - can obey the Eigen equation, and thus have eigen values and eigen functions. - In general we are concerned with the function that obey this equation.

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Classical Mechanics-Position Example Notice that we are using a function of time to describe the position not some fixed value. This function tells you the position at any point in time.

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Classical Mechanics-Position-3D Note that the operator is applied to the position function and the result is the quantity associated with the operator. Ie. The x operator give you the x component of r(t), this is know as a projection operator. The vector operator r can be constructed from the projector operators.

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Classical Mechanics-Position-3D Example

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Classical Mechanics-Velocity-1D Example

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Classical Mechanics-Velocity-3D Example

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Classical Mechanics-Velocity-3D

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The corresponding vector operator to velocity can be reconstructed from the projector operators of the components:

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Classical Mechanics-Velocity-3D Example

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Classical Mechanics-Acceleration-1D Example

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Classical Mechanics-Acceleration-3D Example

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Classical Mechanics-Force-1D Example

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Classical Mechanics-Force-3D Example

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Impulse and Momentum Momentum Impulse In general For a constant force

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Momentum-1D Example

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Momentum-3D Example

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Impulse-1D Force can be thought of as a change in potential energy with change in position

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Impulse-1D Examples i) In terms of the Force operator: ii) In terms of the Potential operator:

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Impulse-3D

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Angular Momentum

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Kinetic Energy

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Potential Energy Hooks Law Coulombs Law

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Conservation of Energy Total energy remains constant, as long as V is not an explicit function of time. (i.e V(x(t))) Hamiltonian

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Conservation of Energy-Hook’s Law Since: Newtons Law F – ma = 0

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