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11.1 The momentum operator as a vector

First of all we know from classical mechanics that velocity and momentum, as well as position, are represented by vectors. Thus we need to represent the momentum operator by a vector of operators as well,

$\displaystyle \hat{\vec{p}}$ = ($\displaystyle {\frac{\hbar}{i}}$$\displaystyle {\frac{\partial}{\partial x}}$,$\displaystyle {\frac{\hbar}{i}}$$\displaystyle {\frac{\partial}{\partial y}}$,$\displaystyle {\frac{\hbar}{i}}$$\displaystyle {\frac{\partial}{\partial z}}$). (11.1)

There exists a special notation for the vector of partial derivatives, which is usually called the gradient, and one writes

$\displaystyle \hat{\vec{p}}$ = $\displaystyle {\frac{\hbar}{i}}$$\displaystyle \nabla$. (11.2)

We now that the energy, and Hamiltonian, can be written in classical mechanics as

E = $ {\frac{1}{2}}$mv2 + V(x) = $\displaystyle {\frac{1}{2m}}$p2 + V(x), (11.3)

where the square of a vector is defined as the sum of the squares of the components,

(v1, v2, v3)2 = v12 + v22 + v32. (11.4)

The Hamiltonian operator in quantum mechanics can now be read of from the classical one,

$\displaystyle \hat{H}$ = $\displaystyle {\frac{1}{2m}}$$\displaystyle \hat{\vec{p}}^{2}_{}$ + V(x) = - $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle \left(\vphantom{\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}}\right.$$\displaystyle {\frac{\partial^2}{\partial x^2}}$ + $\displaystyle {\frac{\partial^2}{\partial y^2}}$ + $\displaystyle {\frac{\partial^2}{\partial z^2}}$ $\displaystyle \left.\vphantom{\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}}\right)$ + V(x). (11.5)

Let me introduce one more piece of notation: the square of the gradient operator is called the Laplacian, and is denoted by $ \Delta$.


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Next: 11.2 Spherical coordinates Up: 11. 3D Schrödinger equation Previous: 11. 3D Schrödinger equation

© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk