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European Journal of Physics (Science) Education Vol.2 No.2 2011 ISSN 1309 7202 Syam and Roy !"#$%&'()*$"#('+)$,)-./0120)!3"2'41$()))))))5$+6)7)8$6)7)))))))))))))))))))79::))))))))))))))))))))))))))));<<8):=9>)?797)) ) @'AB'3&)

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Trajectory of Charged Particle in Combined Electric and Magnetic Fields Using Interactive Spreadsheets

Popat S. Tambade

Prof. Ramkrishna More College, Akurdi Pune 411 044, India

[email protected]

Abstract The objective of this article is to graphically illustrate to the students the physical phenomenon of motion of charged particle under the action of simultaneous electric and magnetic fields by simulating particle motion on a computer. Differential equations of motions are solved analytically and path of particle in three-dimensional space are obtained using interactive spreadsheet. Spreadsheets can be setup to solve numerical solutions of complex systems. This will make the concept, which so far is left to the abstract imagination come alive for the student, and initiate a deeper understanding of particle motion. Keywords: Physics Education, electromagnetic fields, spreadsheets, simulation. Introduction

In recent years, physics educators have started to look more closely at what their students understand about physics concepts. A primary goal of physics education research is to identify difficulties in learning physics through traditional methods and to develop new instructional modes for effective learning (McDermott, 2001). One of the best instructional modes allowing for the development of individual experience is computer simulations. Computer simulations can be the best mode, when the tactile, kinesthetic experience of real objects is impracticable (Arons, 1987). Computer simulations are also useful for providing more extended practice in thinking about a wide variety of examples. It is capable of supplying continual feedback regarding error and correctness and reinforcing the hands on observations when latter have carried out. For proper understanding of physics concepts to the students, teachers can use computer simulations. Teacher should develop computer simulations of physics problems so that it becomes easier for students to understand the phenomena. Spreadsheets can be a powerful tool in physics teaching-learning. From data analysis and graphing to animation and simulations, Microsoft Excel is a very versatile program for the teachers and students. The advantage of using a spreadsheet in teaching-learning process is that the programming is streamlined and less time is needed to enter the necessary code. The strong features of spreadsheet are their cell based structure and the simple interface that is easy to use for new users also. The power and simplicity of a spreadsheet is that the data manipulations are held in front of the user in a very direct and accessible manner. In addition, the spreadsheet program itself provides for screen graphics, charts, and easy-data manipulation using large number of functions, on-screen numerical and visual feedback, and fast calculations (Wagner,

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Quantum-Classical Connection for Hydrogen Atom-like Systems

Debapriyo Syam1 and Arup Roy2

1Department of Physics, Barasat Government College, 10 KNC Road, Barasat, Kolkata - 700124, India

E-mail: [email protected] 2Department of Physics, Scottish Church College

Urquhart Square, Kolkata – 700006, India E-mail: [email protected]

Abstract

The Bohr-Sommerfeld quantum theory specifies the rules of quantization for circular and elliptical orbits for a one-electron hydrogen atom–like system. This article illustrates how a formula connecting the principal quantum number ‘n’ and the length of the major axis of an elliptical orbit may be arrived at starting from the quantum mechanical description and how in the limit when ‘n’ is large one gets the expected classical result. Keywords: Quantum, hydrogen atom, quantum mechanics, one-electron atoms, classical mechanics Introduction According to the non-relativistic version of quantum mechanics, the energy of the electron in a hydrogen-like atom depends only on the principal quantum number n (apart from the atomic number Z of the nucleus.) Again, in non-relativistic classical mechanics, the energy of such a particle, which describes an elliptical path that keeps the nucleus at one focus, depends only on the size (2a) of the major axis. As one expects to recover the classical result in the large n limit of quantum mechanics (‘the correspondence principle’), there ought to be a connection between n and 2a. This connection will be examined in this article. We begin by recalling the salient features of the two approaches: The classical orbits under inverse-square type force (‘Keplerian orbits’): The following quantities remain conserved (Rana and Joag, 1991; Synge and Griffith, 1959): 1. Energy, which depends only on the size of the major axis:

aZeE2

2

!= (1)

2. Angular momentum vector; 3. Runge-Lenz vector, which gives the direction of the major axis. Hydrogen-like atom in quantum mechanics (Ghatak and Lokanathan, 1984): The wave function (ignoring spin) of the system :

),()(),,( !"!" lmnlnlm YrNRr =# (2)


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61

where N is a normalization constant, while

)0(])1(.!21.)1.[(

)!(4)!)(12(.)1(),( 22/2

2/1

!""#$

%&'

(

+

"+"=

+

+

mexdxd

lx

mlmll

Y imlml

ml

lmm

lm)

*)+ (3)

x standing for cos! ; and

),()1(),( ,* !"!" ml

mlm YY ##= (m < )0 (4)

The radial part of the wave function reads

)(..)( 122 !!!

++

"= l

lnl

nl LerR (5) )(12 !+

+llnL being an associated Laguerre polynomial (Ghatak and Lokanathan, 1984) with

)7(2

)6(2

2!

E

r

µ!

!"

=

=

E being the energy (E< 0) and µ the reduced mass of the system. The quantum numbers n, l, m have their usual meanings. Eigenvalue of energy (E) of bound states is given by

22

42

2 neZE

!µ

!= (8)

while

[ ]!

""

=

+++ ++"""

+"=

1

0

2112

!)!12()!1()!(.)1()(

ln

k

kkl

ln kklklnlnL #

# (9)

We are looking for the classical limit of the quantum mechanical solution. This requires

that we take a large value for n. (Also large Z, which, however, is a constant for a given system.) But something else is also to be taken into account. In classical mechanics all three components of angular momentum have well-defined values. There is also a vector (Runge-Lenz vector) which points in the direction of the major axis; we shall come to this later. As the operators that correspond to the components of angular momentum (orbital angular momentum) do not commute with each other, only one component (z-component, by convention) of angular momentum and the magnitude of the angular momentum have, simultaneously, definite values in quantum mechanics. Now, writing the orbital angular momentum as !

"l , we have

)1()ˆ(ˆˆˆ 2222 +==++ llllll zyx (10)


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62

where A stands for the expectation value of the operator A . Now

mlz =ˆ ( - l " m " l ) (11)

with (12) 0ˆˆ == yx ll

If llz ±=ˆ , then 22ˆ llz = , and

(13) ˆˆ 22 lll yx =+

i.e.

( ) ( ) (14) 22 lll yx =!+! , !lx, !ly being the standard deviations of lx, ly ; so that the relative uncertainty in the direction of the angular momentum is small. Conversely, if 0ˆ =zl , then )1(ˆˆ 22 +=+ llll yx ; the

direction of the angular momentum is now totally uncertain. Clearly the classical situation corresponds to llz ±=ˆ . For concreteness we shall take llz =ˆ . As a matter of fact, we shall

consider only large values of l. Notice now that m = zl = l implies

(15) )1(),( 22 llm xY !"#$

and, for large values of l, this quantity is practically equal to zero except when x = 0 or 2/!" = , i.e.on the XY plane (Pauling and Wilson, 1935; Roy, 2010). The electron therefore spends most of its time on the XY plane: the first hint of the emergence of a circular or an elliptical orbit in the classical limit. Connection between ‘n’ and the length of the major axis We shall next find the most probable value of r for the electron; this requires finding the value of r (or the values of r) at which 22 rRnl has a local maximum (maxima.) The cases l = n – 1, n - 2, n – 3, n – 4 can be handled analytically. (Abel’s theorem rules out the possibility of solving equations of degree higher than four by algebraic techniques (Hall and Knight, 1969).

The case l = n-1

Here (16) 222 n

nl erR !!"#


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63

and maximization of 22 rRnl through demanding that ( ) 022 =rRdd

nl!gives n2=! . Using this

value of # (or, rather !"2

=r ) in the classical formula for the energy E (for a circular orbit) viz.

rZeE2

2

!= , we get

this is precisely Bohr’s formula.

Let us work out the width of the distribution of ne 2!!" around the maximum. Putting # = 2n + $, and assuming that n2<<! , we have

(18) )2( 4222

2

nnnn enee!

" "#

## $ Thus the standard deviation, !#, of # is only n2 . The relative width, !! /" , of the distribution becomes smaller with increasing n: The electron is practically confined to a circle on the XY plane for large n ( when m =l and l = n – 1.) It is interesting to calculate the energy associated with radial motion in this case. (One does not expect any from a strictly classical stand point.) There are two ways of calculation:

(i) One may subtract the energy 2

2

2)1(r

llµ!+ , linked to angular motion, from"E"which is just the

total kinetic energy in both classical and quantum mechanics. Setting l =n - 1 and using the

appropriate value of r viz. !!

" nr ==2

we obtain

(19) )(nE

E radial =

(ii) One may, alternatively, use the uncertainty relation to first compute rp! from !r =!"2# ; and

then calculate (E)radial via (E)radial = µ2)( 2rp! . Thus

2

22

)(24)(

!µ"#

$!

radialE # (20) nE

• l = n – 2

(17) 2 22

42

neZE

!µ

!=


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64

The primary aim of this work is to establish the connection between the size of the major axis (2a) and the principal quantum number (n): For a given nucleus (having charge Ze) the energy of the system depends only on ‘a’ in classical mechanics while it depends only on the principal quantum number ‘n’ in quantum mechanics. Remember that the major axis joins the point in the orbit where the radial distance of the electron from the force-centre (nucleus) is least to the point where the radial distance from the nucleus is most; it is also called the apsidal line. As r does not change for small changes in the direction of the radius vector near the major axis, the electron spends relatively longer periods of time at the ends of the apsidal line. Since there is no analogue of the Runge-Lenz vector in quantum mechanics (unless one thinks of superposition of states or deals with wave-packets), distances to the two ends of the major axis may be ascertained by finding the distances to the two maxima of 22 rRnl at which it has the two highest values.


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65

Fig.1. For l = n – 1, 22 rRnl has its maximum value at r = rn = !n . The distribution has a

width of approximately !n2 .

Fig.2. A three-dimensional version of Fig. 1. Fig.3. For l $ n – 2, 22 rRnl has two or more maxima. The values of r at which 22 rRnl has its

two highest magnitudes lie at the ends of the major axis. In fact, these correspond to the largest and the smallest values of r at which 22 rRnl have

its maxima. Note that the number of maxima of 22 rRnl is (n – l). In between two successive maxima of 22 rRnl there is always a minimum of 22 rRnl where the function also goes to zero. Notice that we are relating maximum time spent to maximum of 22 rRnl , presumably a sensible guess. For l = n – 2,

(21) )]22([)!22()( 3222

12 !!!=" !!

++ nnLL n

nlln ##

Thus

22 rRnl %e!![!n ! (2n! 2)!n!1]2 (22)

The maxima of 22 rRnl are located at

(23) ,2234)12( nnnn ±!"±"=±#

while there is a minimum at # = (2n – 2) # 2n. Arguments advanced above lead to a length of the

major axis equal to !""

2)( #+ + #

!n2 . Going back to the expression for the energy in classical

mechanics, we find that such a length of the major axis leads to the correct value of the energy. Thus our expectations are fulfilled for l =n – 2.

• l = n – 3

Here

(24) ]!2)!32()!42()!52(!2

1[])!32[(2

212

!!

!+

!!!"+

+ nnnnL l

ln##


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66

Hence the roots (and minima) of 22 rRnl are at

(25) )32()32( !±!= nn"

To find where the maxima of 22 rRnl lie, let us put !" +#= )22( n . The equation satisfied by & is

(26) 0)1612()1210(3 =!!!! nn ""

Following Cardan [ Hall and Knight, loc cit] let us set & = y + z and require: yz = 3

)1210( !n .

Then

(27) 27

)1210()43(4)43(23

23 !!!+!=

nnny

(28) 27

)1210()43(4)43(23

23 !!!!!=

nnnz .

When n is large

21

6 )4310( !"ney

i#

(1,%, %2) (29a)

and

21

6 )4310( !"

! nezi#

(1,%, %2) (29b)

where (1, %, %2) are the cube roots of unity (1). For large n the required solutions are

)4310(3].[)4

310( 66

1 !=+!"! neen ii ##

$ (30a)


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67

)4310(3]..[)4

310( 266

2 !!=+!"! neen ii

##$%%

(30b)

0)4310( 626

3 =!"

#$%

&+'(

'

))*++ ii

een (30c)

The sum of the largest and the smallest values of # viz. (#1 +#2 ) is approximately equal to

2(2n – 2) or 4n, when n is large. The length of the major axis, by our assumption, is then !n2 and

we are again led to the conclusion that the energy E depends on this particular quantity.

• l = n – 4 In this last algebraically solvable case

]!3)!42(!2)!52()!62(!2)!72(!3

1[])!42[(32

212

!!

!+

!!

!!!"+

+ nnnnnL l

ln### (31)

The minima of 22 rRnl are placed at !" +#= )42( n , where & is a solution of the cubic equation

0)42(2)42(33 =!!!! nn "" (32)

The required values of & can be obtained by Cardan’s method. For large n the solutions are

)42(3].[)42( 661 !=+!"

!

neenii ##

$ (33a)

)42(3]..[)42( 2662 !!=+!"

!

neenii

##$%%

(33b)

0)42( 6263 =!

"

#$%

&+'(

'

))*++ ii

een (33c)


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68

To find the values of # where 22 rRnl has its maxima, let us put # = (2n-3) + & . Some algebraic manipulations lead to the following equation for& :

(34) 0)15312624()95(8)53(6 224 =+!+!!!! nnnn """

Collecting the leading terms (remember that we are interested in the large n situation) we have

0244018 224 =+!! nnn """ (35)

This equation can be solved by the technique invented by Descartes [Hall and Knight, loc cit]. For pedagogical reasons we give a brief outline of the method. Consider the equation

(36) 024 =+++ srxqxx

Assume

(37) ))(( 2224 mkxxlkxxsrxqxx +!++=+++

Then by equating coefficients we get

(38) 0)4(2 22246 =!!++ rksqqkk

This is a cubic equation in k2. After solving this equation we can find the values of m and l. Finally x (four possible values) is obtained from the pair of equations

02 =++ lkxx (39)

and 02 =+! mkxx . (40)

Applying this technique to our equation for &, we get

2

)5

195(42828,

2

)5

195(42828 nnnnnnnn +!±!!!±"# (41)


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69

In the large n situation the highest and the lowest values of & are respectively n)828(21

+

and n)828(21

+! . Clearly the sum of the corresponding values of # is approximately equal

to 4n, as anticipated. Our stand is therefore vindicated.


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70

Fig.4. Plots (schematic) of the variation of 22 rRnl with # for different values of l. Conclusions The cases for other ‘l ’values may be handled by numerical methods. However, the trend is evident, for it has to be consistent with Bohr’s correspondence principle. We hope that through this article we have been able to illustrate how quantum effects disappear and classical results emerge in the limit of large ‘n’ values. It may also help the students to accept quantum mechanics as a deeper and refined description of nature.

References

Ghatak, A. K. and Lokanathan, S. (1984). Quantum Mechanics – Theory and Applications, 3rd

eds. McMillan India Ltd. Hall, H. S. and Knight, S. R. (1969). Higher Algebra. McMillan and Co. Limited: London.

Pauling, L. and Wilson, E. B. (1935). Introduction to Quantum Mechanics, Intl. Student ed.

McGraw-Hill- Kogakusha Book Co., Inc./ Kogakusha Company, Ltd.: Tokyo. Rana, N. C. and Joag, P. S. (1991). Classical Mechanics. Tata-McGraw Hill Publishing

Company Limited. Roy, A. (2010), Introduction to Quantum Mechanics (Miley-Mishey) Synge, J. L. and Griffith, B. A. (1959). Principles of Mechanics, 3rd ed. /Intl.Student ed. McGraw

Hill Book Company.


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quantum-classical connection for hydrogen atom-like ... · pdf filepopat s. tambade prof....

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