Mathematica statements

Mathematica statements can contain symbols for stored variables as well as constants. A variable stands for a location in the computer memory in which a numerical value can be stored. Variable names can contain any number of letters and/or digits. However, they cannot begin with a digit. Begin your variable names with a lowercase letter to avoid confusion with Mathematica functions and other Mathematica objects, which always begin with a capital letter. Also remember that xy would represent a variable called xy while x y (with a space between the letters) stands for the product of the two variables x and y. 

Mathematica has a powerful capability to carry out symbolic mathematics on algebraic expressions and can solve equations symbolically. In addition to the arithmetic operations, the principal Mathematica statements for manipulating algebraic expressions are Expand ], Factor ], Simplify ], Together ], and Apart ]. The Expand statement multiplies factors and powers out to give an expanded form of the expression. The following input and output illustrate this action In l] =Clear a,x]... 

Mathematica statements can contain symbols for stored variables as well as constants. A variable stands for a location in the computer memory in which a numerical value can be stored. Variable names can contain any number of letters and/or digits. However, they cannot begin with a... 

The absorption spectrum consists of sequences of transitions from v" = 0, 1, 2 to various v levels in the upper state, and the relative intensities of the vibration-rotation bands are given primarily by the product of the FCF value and a Boltzmann term, which can be taken to be exp — hcv v /kT). Common choices for the i/r s are harmonic oscillator and Morse wavefunctions, whose mathematical form can be found in Refs. 7 and 9 and in other books on quantum mechanics. The harmonic oscillator wavefunctions are defined in terms of the Hermite functions, while the Morse counterparts are usually written in terms of hypergeometric or Laguerre functions. All three types of functions are polynomial series defined with a single statement in Mathematica, and they can be easily manipulated even though they become quite complicated for higher v values. 

A Mathematica calculation of Franck-Condon factors that determine electronic transition intensities of I2 is presented in Chapter III, and program statements for this are illustrated for I2 in Fig. III-6. In this fignre, note the dramatic differences between the intensity patterns predicted for the harmonic oscillator and Morse cases and compare these patterns with those seen in your absorption spectra. If yon have access to this software, yon might examine the changes in the harmonic-oscillator and Morse-oscillator wavefnnctions for different v, v" choices. A calcnlation of the relative emission intensities from the v = 25, 40, or 43 level conld also be done for comparison with emission spectra obtained with a mercury lamp or with a krypton- or argon-ion laser, hi contrast to the smooth variation in the intensity factors seen in the absorption spectra, wide variations are observed in relative emission to v" odd and even valnes, and this can be contrasted with the calcnlated intensities. Note that, if accnrate relative comparisons are to be made with experimental intensities, the theoretical intensity factor from the Mathematica program for each transition of wavennmber valne v shonld be mnltiphed by v for absorption and for emission. ... 

Most of the Mathematica symbols are the same as those used in Excel or various computer programming languages such as BASIC except for the use of a blank space for multiplication. Excel and BASIC use only the asterisk for multiplication. In ordinary formulas, placing two symbols together without a space between them can stand for multiplication. In Mathematica, if you write xy, the software will think you mean a variable called xy, and not the product of x and y. However, you can write either 2x or 2 x for 2 times x, but not x2. It is probably best to use the asterisk ( ) for multiplication rather than a space in input statements. Watch for the use of the blank space in output statements. Complex arithmetic is done automatically, using the capital letter I for. Several constants are available by using symbols Pi, E, I, Infinity, and Degree stand for n,e,i =, oo, and... 

The underseore following the symbol for the independent variable in the fune-tion expression on the left-hand side of the statement is part of the function definition and must be typed in. Mathematica s second type of equal sign, = (a colon followed by an equal sign), must be used. After defining a function, you can use it in a Mathematica expression, as in the statement g=x f[x] Cos[x/y]... 

The Clear statement is included in case a and x had been previously defined as variables with specific values, which would cause Mathematica to return a numerical result instead of a symbolic result. 

The trial value of the root is here represented by the 1 following the x in braces (curly parentheses, ). If your equation is a polynomial equation, the NSolve statement can be used instead of FindRoot. The NSolve statement does not require a trial root, and will find all roots, while the FindRoot statement will generally cause Mathematica to converge to one root and then stop. If your equation has more than one root, you need to determine whether you have found the desired root and not one of the others. 

The Apart statement in Mathematica carries out the decomposition into partial fractions automatically. See Chapter 3. 

Mathematica can carry out indefinite integrals symbolically. For example, the input and output statements for the indefinite integral of sin(x) are In[l] =Clear[x]... 

Mathematica carries out numerical solutions of differential equation for which no exact solution can be written. The solution is given in terms of an interpolating function, which is a table of values of the unknown function for different values of the independent variable. The program finds a numerical value of the function for a specific value of the independent variable by interpolation in this table. The statement NDSolve is used to solve the differential equation, as in the next example ... 

As you can see, matrix multiplication with fairly large matrices can involve a lot of computation. Computer programs can be written to carry out the process, and such programs are built into Mathematica and also into computer languages such as BASIC so that a matrix multiplication can be carried out with a single statement. 

Note that the symbol that we chose for the matrix name is in lower case and requires no auxiliary labels. You should start the names of all Mathematica variables with lowercase letters to avoid possible confusion with Mathematica operators and functions. If you want to see the matrix A in standard form, type the statement MatrixForm[a] and press the Enter key or the Shift-Return. ... 

If the matrix is singular and does not have an inverse, Mathematica will tell you so. To obtain the determine of a square matrix A, use the statement Det[a]... 

Remember the capitalization. Mathematica does not allow alternate choices to its statements. 

In Chapter 3, we introduced the use of Mathematica to solve a single algebraic equation, using the Solve statement and the NSolve statement. The Solve statement can also be used to solve simultaneous equations. The equations are typed inside curly brackets with commas between them, and the variables are listed inside curly brackets. To solve the equations ax + by = c gx + hy = k we type the input entry... 

Mathematica finds matrix eigenvalues and eigenvectors by use of the statements Eigenvalues[m] and Eigenvectors[m], where m denotes a matrix that has already been typed into the program. 

SOLUTION We open Mathematica and type the input statement... 

Therefore, the best bet for us it to use the analytical solution to solve for h[t] numerically or graphically. The graphical method is not used much today but it is worth illustrating because it reinforces a good "feel" for the functions and the numbers that result. We will look at the graphical solution first. As you no doubt recall, the method used finds the solutions to this equation that make the left- and right-hand sides equal. We can view each side as a statement for two different functions that intersect at certain points these intersections are the solutions. If we graph the two functions we can find these points. This is very easy to do in Mathematica as follows. 

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