MSc-PRACTICAL EXPERIMENTS

MSc-PRACTICAL EXPERIMENTSMSc-1- FOURIER SERIES MSc-2-BOHR'S THEORY OF ATOMIC STRUCTURE

MSc-3-NUMERICAL SOLUTION OF SCHRODINGER WAVE EQUATIONMSc-4-LORENTZ FORCE

MSc-5-EULER'S METHOD OF SOLVING ODE

MSc-6-NEWTON RAPHSON METHODMSc-7-NUMERICAL INTEGRATION-3-8 SIMPSON RULE

MSc-8-RANGE-KUTTA METHOD -ODE-MSc-9-NUMERICAL SOLUTION OF SECOND ORDER DIFFERENTIAL EQUATIONMSc-10-HARMONIC OSCILLATOR

MSc-11-LEARNING MOLECULAR SYMMETRY-H2O-NH4MSc-12-STUDY MATERIAL OF HYDROGEN ATOM AND MOLECULEMSc-13-GROUND STATE OF HELIUM ATOM -VARIATION METHODMSc-14-ELECTRON MOTION THROUGH RECTANGULAR POTENTIAL BARRIER

MSc-15-NUMERICAL SOLUTION FOR THE RADIAL EQUATION OF THE HYDROGEN ATOMMSc-16-NUMERICAL SOLUTION OF THE RADIAL EQUATION FOR 3D OSCILLATORMSc-17-XY3- PLANAR- H2CO-FORMALDEHYDE.MSc-18-ETELON1. FOURIER SERIES

Goal of this exercise:

To give students the opportunity to efficiently and effectively practice using a Fourier Series Expansion. This exercise serves as a foundation for future work in quantum mechanics where linear combinations of atomic orbitals and expressions of solutions to the Schrodinger Equation are written as expansions using sets of functions that are complete orthonormal sets.FOURIER SERIESThe general equations for a fourier series are shown here. Notice the use of a and b for the coefficients for the cos and sin terms respectively. Here F(x) is the function that is to be fit by the Fourier series. In the sections that follow you will be lead through a series of exercises that will put into practice the Fourier series method for both odd and even functions.

regular functions, each continuous component of the function must be determined These expressions for a and b assume that the integrals exist and that F(x) is a continuous function. For piecewise separately.MATHCAD WORKSHEET WORKLet us start with a simple Fourier expansion for a periodic function, a simple step function. When you work with this document do not repeat the exercises using the same variable names. This will confuse Mathcad and give you a headache trying to debug your Mathcad code.

Vary N, the number of terms in the expansion and describe the consequences as N is increased to a large number. Record your observations in your notebook. Note how the Fourier series behaves relative to the step function it is designed to match.

Concisely express the function that is being expressed by the series expansion shown above. This means to write out explicitly the first few terms in the series for the fitting function.Sample Problem:

Expand the function F(x)=x in the interval - to by a Fourier series.Solution: This function is an odd function. Only sine terms will show up in the expansion.

2. BOHR'S THEORY OF ATOMIC STRUCTURE

Postulates of Bohr:

1. An electron in atom can revolve in certain stable orbits without the emission of radiant energy.The angular momentum of the electron should in the integral multiples of (nh/2)

2. An electron may make a transition from one of its specified non-radiating orbits to another of lower energy equal to the energy difference between initial and final states.

In stationary states the energy of the atom is assumed to be constant. Thus the atom does not radiate em waves while in any one of these stationary states.

Theoretical Spectrums of the hydrogen atomLYMAN SERIES

BALMER SERIES

PASCHEN SERIES

BRACKET SERIES

FUND SERIES

3. NEWTON-RAPHSON METHODWhen a fluid flows in a steady fashion through a round pipe, the pressure drop due to wall friction is given by the empirical formula:

Several empirical formulas exist for the friction coefficient as a function of the Reynolds number, Re, which is given by where is the fluid viscosity. For flow in the turbulent regime between completely smooth pipe and wholly rough pipe surfaces Colebrook developed an empirical equation for the friction factor

Develop a procedure to determine f for specified values of /D and Re. Use the approximation proposed by Genereaux (1939) , , to determine an initial approximation for f. The equation is transcendental in anture therefore an analytical solution is not possible. Here we will use the Newton Raphson procedure. A graphical description of the procedure is given below.SOLUTION - find the friction factor using the Newton Raphson procedureDefine values for /D and the Reynolds Number

The derivative w.r.t. f of the Colebrook White equation

4. EULER'S METHOD FOR SOLVING ORDINARY LINEAR DIFFERENTIAL EQUATIONSIt is often impossible to find elementary solutions to differential equations of the form y' = f(x,y). In applications, though, it is usually sufficient to find a numerical approximation to the function satisfying the governing differential equation. We will develop a simple numerical technique for the estimation of such solutions.

Note that a derivative "function" can be denoted in the same way as the original function from which it came.

In order to obtain the required relationship between variables we need a starting point, commonly called an initial condition: , . This is the initial condition for the desired realtionship between x and y.

We will use this function to rate the accuracy of our estimates. Each method will rely on iteratively

estimating values at N+1 evenly spaced points on [0,L]. Here, we take

5. RUNGE KUTTA METHOD

A Second Order Runge Kutta MethodThe simplest method works for differential equations of the form y' = f(x,y): Note: we are operating on a function which specifies the derivative of another function. Our solution will be the functional relationship between x and y We must also have an intial condition, namely:

6. NUMERICAL INTEGRATION 3/8-SIMPSON'S RULE

Stream cross sectional areas (A) are required for a number of tasks in water resource engineering, including flood forecasting and reservoir design. Unless electronic sounding devices are available to obtain continuous profiles of the channel bottom, the engineer must rely discrete depth measurements to compute A. An example of a typical stream cross section is shown below. The data represents places where a boat was anchored and depth measurements taken. Estimate the cross sectional area of the channel.

Data below scaled from drawing

In Mathcad origin default is 0, reset to 1 in the Math / built-in variables submenu

Disatance depth

compute the flow area using Simpson's rule

Simpson's 3/8 rule - each segment requires 4 points

If the average flow velocity in the stream is 6 fps what is the flow rate?

7. LORENTZ FORCE

We study the motion of a particle of charge q and mass m in a magnetic field B and in an electric field E. The force acting on the particle is given by the Lorentz formula: F = qE + qvxB where v is the velocity vector. For small enough velocities, v