### Description

PH221 – Week 7 Lab

# Wave Functions

Welcome to this week’s laboratory component. This week is different than previous weeks in that you will not be using a KET simulation for this activity. Instead, you will be creating and analyzing graphs of a wave function using Microsoft Excel.

Download the instructions for the laboratory activity you will complete this week. You may wish to print it out and use it to collect and organize your results.

Use the answers to the laboratory questions to help you write your lab report. Your report will focus on graphs of the wave function vs. position. You should discuss what these graphs mean in terms of probability of finding a particle at specific position at a point in time.

The lab report will have the following six sections. Include section headings in bold at the beginning of each section.

1.Introduction – Explain the purpose of this laboratory and what results you expect to see in this experiment.

2.Background – Discuss the concepts that form the foundation for this lab. You should address what you learned from the weekly lectures and readings that are related to the lab.

3.Methodology – Describe the apparatus that was used in the experiment(s) and how it was used in performing the experiments. Also explain what tools were available within the laboratory that allowed you to collect or analyze the data.

4.Data – Enter the data that you collected in the lab. You can use screen shots from the Data Table within the Pivot Interactives labs. Data should be clearly labeled with physical quantities and units.

5.Analysis – Analyze your results. If your Data Table included Calculated Columns, then the equation you used in those calculations should be included and described here. Any graphs created with the data go in this section, as well as your interpretations of their meaning. Were your results consistent with your original expectations?

6.Conclusion – Provide a concise summary of the results of your experiment(s) – what you did, what you found and what it means. Speculate on possible sources of experimental error and/or uncertainty within the experiment. Describe an additional experiment that could be run with this equipment to expand on what you’ve learned OR explain how you could use this equipment to answer another real-world problem.

## Explanation & Answer

Attached.

Lab 7 Wave Functions

Introduction

This lab was conducted to study about wave functions and their significance in determining the

various properties in physics. It was meant to analyze the wave function of the infinite square well,

2

𝑛𝜋𝑥

).

𝐿

expressed as 𝜓𝑛 (𝑥) = √𝐿 sin(

Background

A wave function is a function that describes the probability of quantum state of a particle as a

function of time, position, spin or momentum. The wave function is mostly denoted as Ψ. In

particular, a wave function can be used to estimate the probability of finding an electron

within a matter wave. A wave function can either be a real or imaginary number. In cases of

an imaginary number, the function is squared to generate a real number. A plot of these

numbers may generate a wave form with the amplitudes as the yield of the function. The

probability of finding an electron in a certain area can the be assessed from this function and

is proportional to the amplitude of the wave.

Methodology

2

𝐿

𝑛𝜋𝑥

).

𝐿

The excel program was used to generate data based on a wave function 𝜓𝑛 (𝑥) = √ sin(

The

values of x were taken between the range 0 to 1 with a common difference 0f 0.05. The value of L

was set as 1. For every value of x, the wave function was determined. Also, a square of the wave

function was also determined. These values were done for n = 1, 2, and three and the values in put

in three tables. These data were then used to plot a total of six graphs with two graphs for each

value of n, that is, Ψn(x) vs. x and Ψn(x)2 vs. x.

Data

Table 1: Wave Functions for n = 1

x

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

n=1

Ψn(x)

0

0.559383

0.786201

0.952938

1.084298

1.189268

1.272076

1.334969

1.379208

1.405501

1.414214

1.405456

1.379116

1.334831

Ψn(x)2

0

0.312909

0.618111

0.90809

1.175702

1.414358

1.618178

1.782142

1.902214

1.975434

2

1.975307

1.901962

1.781773

0.7

0.75

0.8

0.85

0.9

0.95

1

1.271888

1.189025

1.083994

0.952557

0.785708

0.558663

0

1.617699

1.413781

1.175043

0.907364

0.617337

0.312104

0

Table 2: Wave Functions for n = 2

n=2

x

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Ψn(x)

0

0.786201

1.084298

1.272076

1.379208

1.414214

1.379116

1.271888

1.083994

0.785708

0

0.786696i

1.084601i

1.272266i

1.379301i

1.414214i

1.379025i

1.271700i

1.083688i

0.785213i

0

Ψn(x)2

0

0.618111

1.175702

1.618178

1.902214

2

1.901962

1.617699

1.175043

0.617337

0

-0.61889

-1.17636

-1.61866

-1.90247

-2

-1.90171

-1.61722

-1.17438

-0.61656

0

Table 3: Wave Functions for n = 3

n=3

x

0

0.05

0.1

0.15

0.2

0.25

0.3

Ψn(x)

0

0.952938

1.272076

1.405501

1.379116

1.189025...