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Symbolic Math Approach to Solve Particle-in-the-Boxand H-atom Problems

Todd M. Hamilton, Ph.DDepartment of ChemistryGeorgetown CollegeGeorgetown, KY 40324

[email protected]

Copyright 2006 by the Division of Chemical Education, Inc., American Chemical Society. All rights reserved. For

classroom use by teachers, one copy per student in the class may be made free of charge. Write to JCE Online,

[email protected], for permission to place a document, free of charge, on a class Intranet .

Goal

This exercise is designed to give undergraduate physical chemistry students a "hands-on"

experience in solving the hydrogen atom. The students explore a step-by-step solution to theSchrdinger equation in a symbolic way and explore properties of the radial wavefunction.

Objectives

Upon completion of the exercise students should be able to:

1. solve for the energy of the particle in a box symbolically

2. solve the Schrdinger equation symbolically for the energy of the hydrogen atom

3. explain how a wavefunction is normalized

4. discuss the features of the 1s radial wavefunction for the hydrogen atom

5. explain how the size of the 1s orbital is determined

6. solve for the energy of the one-dimensional harmonic oscillator symbolically

Prerequisites

This exercise is intended for junior-level students who have been introduced to quantum

mechanics, particularly the particle-in-a-box problem. Two semesters of calculus and a

semester of physics is desired. Students should also have some experience working with

derivatives and integrals in Mathcad.

References

1. Levine, I. N. Quantum Chemistry, 5th Edition, Prentice Hall, 2000.

2. Levine, I. N. Physical Chemistry, 5th Edition, McGraw-Hill, 2002.

3. Hanson, D. M. and Zielinski, T. J. Quantum States of Atoms and Molecules, CD-ROM, 2001.

4. Newhouse, P. F. and McGill, K. C. J. Chem. Educ.81, 424 (2004).

5. Mak, T. C. W. and Li, W. J. Chem. Educ. 77, 490 (2000).

6. Turner, D. E. J. Chem. Educ. 70, A185 (1993).

7. Lain, L., Toree, A., and Alvarino, J. M. J. Chem. Educ. 58, 617 (1981).

8. Peterson, C. J. Chem. Educ. 52, 92 (1975).

Created March 2005Updated August 2005Updated February 2006

H_AtomStudent.mcdTodd M. Hamilton

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Introduction

The Schrdinger equation can be solved exactly for the hydrogen atom. After some practice with

the particle in a box problem, you will solve the H atom symbolically using the derivatives feature in

MathCad. The result is an expression for the energy levels of the hydrogen atom. You will also

prove that the radial part of the wavefunction is normalized and explore the concept of an orbital.

The Schrdinger equation in spherical coordinates for the H atom is as follows:

h2

2 r2

rr2

rd

d

d

d

1

sin

sin

d

d

d

d+

1

sin2

2

d

d

2+

Z e2

r

= E

where h is actually h-bar (h divided by 2 ), is the wavefunction, is the reduced mass, Z isthe atomic number, e is actually e-prime (e divided by sqrt(4 0)), and E is the function for

the allowed energy levels. The e2

term is the Coulomb potential energy function.

After a separation of variables, the radial part of the Schrdinger equation can be set equal to

a constant, :

h2

R r( ) rr2

rR r( )

d

d

d

d

2 r2

R r( )E

Z e2

r

+

R r( )+ =

where R(r) is the radial wavefunction and = ( +1). For an s orbital, =0 and =0. After setting

= 0 and a little algebra, the Schrdinger equation becomes:

rr2

rR r( )

d

d

d

d

2 r2

h2

Z e2

r

R r( )+ =

2 r2

h2

E R r( )

After defining the radial wavefunction R(r), you will perform the derivatives in the first part

of the equation. Then you will add in the potential energy term. Finally, you will isolate the

energy function. In this exercise, we are focusing only on the radial wavefunction for the 1s

orbital (n=1) in order to simplify the mathematics.



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A. Practice with the Particle in a Box

Before you tackle the H atom, you will obtain some practice solving the particle in a box (PIB).

The wavefunction for the PIB is as follows:

2

A

12

sin n x

A

:= n

The Schrdinger equation for the PIB is very simple (with the potential energy V=0):

2x

d

d

2= 2 m

E

h

2

2

where m is the mass of the particle, h is Planck's constant, and E is the allowed energies ofthe particle. Begin by taking the derivative of the wavefunction (select the expression below

and hit Ctrl+period, then enter). You should get a cosine function as the result.

xd

d

Now try the second derivative of the wavefunction below. You should get the original

wavefunction back (in fact this a requirement for solving the Schrdinger equation) times

some constants.

2x

d

d

2

Now, divide the result from the second derivative by -2m /(h/2)2 (see the Schrdingerequation above) to isolate the E function. To do this, copy and paste your latest result into

the little black box below (or retype the result). Click outside of the math area, then backinside, hit Ctrl +period, then enter:



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2 m

h

2

2

The answer should be E=n2h2/(8ma2)

Since the particle has a 100% probability of being found somewhere in the box, that is with an x

value somewhere between 0 and a, the integral of the probability density function 2betweenx= 0 and x = A must be equal to 1. To assure this we write the wave function for the particle as

(x) = N sin(nx/A), where N is called the normalization constant, and then determine the valueof N that will make

0

A

x

2

x( )

d 1=

We will use Mathcad's ability to solve equations symbolically to to determine the value of the

normalization constant. Below is the equation for the integral of2with written out asN sin(nx/A).

0

A

xN sinn x

A

2

d 1=

Copy this equation into the space below. Important: Since n must be a positive integer, replace

n with any positive integer before completing the next steps.

Next:

1. Click in the space between the left parenthesis and the N so that N and only N is enclosed in

the blue selection L.

2. Go to the symbolics menu on the tool bar, click on variable, then click on solve.

This tells Mathcad to solve for the value of N that will make the integral equal to 1. Notice that

you get two solutions, one positive and one negative. Only the positive value is meaningful.



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Replace your original choice for n with another integer and repeat steps 1 and 2. Notice that you

get exactly the same solution(s) for N. For the particle in a box the normalization constant is the

same for all states (all values of n). Can you simplify the expression Mathcad gives for N into a

more familiar form?

The PIB wavefunction is normalized. In other words, the integral of2 over the boxlength a is equal to 1 (see below). Change the upper limit of integration by changing

the variable "factor" below (try 2 for half the box, 3 for one-third of the box, etc.). After

changing the upper limit, try different n values by changing the n value below (try 2, 3, 4,

etc.) Also, notice how the graphs below change for different n values.

factor 1:= A 1:= n 1:= cn

A:=



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0

A

factor

x2

A

sin c x( ) sin c x( )

d 1=

0 0.5 12

0

2

2

A

1

2

sin c x( )

x

=

0 0.5 10

1

2

2

A

sin c x( ) sin c x( )

x

2 =

B. The Solution to the Hydrogen Atom

The general form of the radial wavefunction is a byproduct of solving the Schrdingerequation. We will restrict ourselves to the 1s orbital to simplify the mathematics. The

radial wavefunction for the 1s orbital in the hydrogen atom is R(r) = 2(Z/a) 3/2e-Zr/a.where a is a constant with dimensions of length (a = 0.5295 ).

1. Type in the wavefunction in the space below.



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R:=

2. Take the first derivative of the wavefunction with respect to r (copy and paste in the

wavefunction that you typed in above):

r

d

d

3. Next, multiply the result by r2 and take the derivative again.

rr2( )d

d

4. Now you are ready to add in the potential energy term. The full term is:

2 r2

h2

Z e2

r

R r( )

where Ze2/r is the Coulomb potential energy function.



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However, the symbolic solution is easier to find if we combine some terms. For example, if

we note that the constant a = h2

/e2

, we can rewrite the above term as2 Z r R

a . Add this

term to the latest result and evaluate. If you have trouble copying and pasting the result to

the right of the evaluation arrow, copy the terms to the left of the arrow instead.

2 Z r R

a+

5. As shown above in the Introduction, the current result is equal to

2 r2

h2

E R r( )

Divide your result by



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2 r2

R r( )( )

h2

to obtain E, the energy function for the 1s orbital in the hydrogen atom.

2 r2

R

h2

Notice that this function contains only one parameter - the atomic number, Z - and is

applicable to hydrogen-like atoms as well. For the H atom, the ground state energy

level is -13.6 eV. The n=2 level requires a different (more sophisticated) wave functionto solve the Schrdinger equation.

In the calculation of the 1s energy level below, change the Z value from 1 (for H) to a higher atomic

number to calculate the energy levels for hydrogen-like atoms. Try uranium!

Z 1:=



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J/eV (conversion factor from Joules to electron volts)

angstroms

J

kg

eV

JpereV 1.602177 1019

:=

a 0.529466 1010

:=

h6.62608

2 3.141610

34:=

9.10443 1031

:=

E =1

2

Z2

a2

h2

JpereV 13.598=

C. Normalization of the H Atom Wavefunction

You will now prove that the radial wavefunction for the 1s orbital in the hydrogen atom is

normalized. First, square the radial wavefunction. The wavefunction is provided below to aid

in the process; type in the expression for the squared wavefunction.



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Z 1:=

R 2Z

a

3

2

expZ r

a :=r

R2

:=R2

Next, multiply the squared wavefunction by r2 and integrate over dr. The factor of r2 takes into

account the fact that you are integrating the volume inside a surface located at a distance r

from the nucleus. Note that you are integrating from 0 to infinity. Again, if you have trouble

with copying and pasting, simply enter "R2" into the integral below.

0

rr2

d

Examine the results. Is the radial wavefunction for the H atom normalized?

D. Plotting the H Atom Wavefunction

Plot the radial wavefunction R versus r for Z=1 (copy and paste the radial wavefunction

R into the box on the y-axis). Set the y-axis limits to (0,2) and the x-axis limits to

(0,5).



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a 1:=

Z 1:=

Now plot the square of the radial wavefunction R2, times r2, versus r (put the r2 factor

at the end of the R2 equation). Set the y-axis limits to (0,0.6) and the x-axis limits to

(0,5).

E. Orbitals

The integral of the function that you just plotted is equal to the probability of finding the

electron at a particular distance from the nucleus. Orbitals illustrated in textbooks usually

define the surface of the orbital at the distance r which contains 90% of the electron

probability. In the active region below, the upper limit is in multiples of the constant a

(remember a is approximately 0.53 ). You see that a sphere of radius equal to theconstant a contains about 32% of the probablility. Change the upper limit of integration



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until the sphere contains 90% of the probability (until the integral evaluates to exactly 0.9).

a 1:=

Z 1:=

upperlimit 1:=

0

upperlimit

r4Z

a

3

e

2 Z ra

r2

d 0.323=

The upper limit value that you just found is the radius (as a multiple of a) that contains

90% of the electron probability. This is the size of the 1s orbital usually displayed intextbooks. Find approximately where this radius is on the plots above (in Section D).

You have defined the surface of the 1s orbital of the H atom.

How far out must you integrate to get essentially 100% of the probability (until the integral

evalutes to exactly 1)?

F. Mastery Exercise

The Schrdinger equation for the One-Dimensional Harmonic Oscillator is as follows:

2x

d

d

2+ 2 m E h

2

2x

2( ) = 0

where = 2m/h, h is actually h-bar, and m is the mass of single particle attracted toward theorigin by a force proportional to the particle's displacement from the origin:



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Fx

= -kx

The force constant, k, is related to the vibrational frequency, , as follows:

= (1/2)(k/m)1/2

The first two wavefunctions that solve the Schrdinger equation are:

0

=

1

4

e

x

2

2

1

=4

3

1

4

x e

x2

2

Show that these wavefunctions solve the Schrdinger equation for the One-Dimensional

Harmonic Oscillator and state the energies of the first two levels. You will rearrange the

Schrdinger equation to solve for the energy (similar to what was done for the H atom).

Important note: open a new window to solve this section.

Also, plot the wavefunctions and prove that they are normalized (set = 1 for the purposes ofplotting and normalization). Remember to integrate from - to + .



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