Preliminary comments

To define a constant, use a single equals sign:

Once defined, you can use the constant anywhere you want:

To avoid problems,  erase from memory  the assigned value to a

When defining a function, you need to remember two important things:
    Use an underscore character after each argument name on the left-hand side (but not on the right-hand side)
    Use a := in the middle

Once defined, you can use the function

The square root of 10

We need to tell MATHEMATICA that we need a numerical value.  We can use a decimal point

or we can use the N function and ask for 50 digits:

N[ Sqrt[10], 50 ]

3^100

N[%]

The command % refers to the previous output

In the following expression I stand for

(3 + 4 I) ^10

More complicated functions

Notice the equal ":=" sign that implies a delay calculation, and the underscore "_" sign that implies the independent variable.

Plot a function

Make sure that you do not mix the "( ), { } and [ ]" brackets.  MATHEMATICA gets confused and it will send you an error message.

Find the zeros

Notices "==" logical sign that determines equality.

We can take the derivative of the function:

or

We have to tell MATHEMATICA to erase from memory defined constants or functions

or

Integration

in1=NIntegrate [ Sin [Sin[x]], {x, 0, Pi} ]

or

Relevant integrals in the case of the particle in the box.  Here we use regular integration and also we considered a useful mathematical technique to find integrals.

We need to consider some substitutions using the "/." command

Notice that "/. tells MATHEMATICA to substitute and that the actual substitution is given by  the arrow command ->"and multiple substitution are enclosed by a curly bracket and separated by commas.

Finally consider the following integral:

A relevant application in Quantum Mechanics

First we consider the integral of the square of the wave function

Since α is a positive real number the integral is equal to unity.  Thus we say that psiH10 is normalized.

Now we take the  derivative of the wave function

The average <> is proportional by the following integral:

Problem 1. - Find the value of <> .