Operator del squared

The Laplacian54, or del-squared, operator is V V = V2 (sometimes shown as A to further confuse the poor reader) V V is given in Cartesian space as... 

Because the del-squared operator acts only on position coordinates, both it and the potential energy term on the left-hand side of Equation (3.43) are independent of time and the (p(t)s cancel on this side. Likewise, because the operator on the right-hand side of Equation (3.43) is independent of the coordinates, the y/ x, y, z)s cancel on the right-hand side. Setting both sides equal to the separation constant (a scalar quantity) yields Equation (3.44). 

The second derivative over each coordinate is called the del-squared operator . The motion of the particle is not separable in Cartesian... 

The del-squared operator must now be converted to spherical coordinates. 

The differential operator on the left side of the equation is known as del-squared. The operator del is equivalent to partial differentiation with respect to x, y, and z components ... 

This is tlic equation of wave motion. V (read as del squared ) is the Laplacian operator and represents... 

The operator V V occurs so commonly that it has its own name, the Laplacian operator and its own symbol, V, sometimes called del squared. It is an operator that occurs in the Schrodinger equation of quantum mechanics and in electrostatics. 

Ihe operator in parentheses in (3.45) is called the Laplacian operator (read as del squared ) ... 

Generalizing from one-dimension to three-dimensions and substituting del-squared as the operator yields Equation (3.41), the time-dependent Schrodinger equation in three-dimensions, which is the same as Equation (3.33). 

The Laplacian operator is the dot product of V with itself. It is w-ritten V- and pronounced del squared. For Cartesian coordinates, the Laplacian operator is... 

In order to properly write the complete form of the Schrodinger equation for helium, it is important to understand the sources of the kinetic and potential energy in the atom. Assuming only electronic motion with respect to a motionless nucleus, kinetic energy comes from the motion of the two electrons. It is assumed that the kinetic energy part of the Hamiltonian operator is the same for the two electrons and that the total kinetic energy is the sum of the two individual parts. To simplify the Hamiltonian, we will use the symbol V, called del-squared, to indicate the three-dimensional second derivative operator ... 

This definition makes the Schrodinger equation look less complicated. is also called the Laplacian operator. It is important to remember, however, that del-squared represents a sum of three separate derivatives. The kinetic energy part of the Hamiltonian can be written as... 

The first term in Equation (A9.2) is the kinetic energy operator, which contains the Laplacian ( del squared ). This is the three-dimensional equivalent to the onedimensional kinetic energy operator we met for the harmonic oscillator in Appendix 6. In Cartesian coordinates, the Laplacian operator is defined as... 

This operator is related to the Laplacian operator, (del squared), in spherical polar coordinates. From the definition of in Cartesian coordinates, and from the coordinate transformation given previously. 

The scalar operator (read del squared ) is known as the Laplacian operator ... 

Here N plays the role of an imaginary time, the Laplacian operator 7 (del or nabla squared) acts upon the variable r and P is the mean square length of a segment. Since a short chain must have its ends in close proximity to one another, the appropriate boundary condition is... 

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