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Unified Representation of Quantum Mechanics on

One-dimensional Harmonic Oscillator

Yongqin Wang

Department of Physics, Nanjing University, Nanjing 210008, China

AbstractA quantum state corresponds to a specific wave function. We adopt a new

mathematical method [1] to improve Diracs ladder operator method. A set of

orthonormal wave functions will be used to associate the operator with the square

matrix corresponding to it. These allow us to determine the matrix elements by using

the operator relations without having to know the specific wave functions. As a result,

we can get the direct results of matrix mechanics and wave mechanics on

one-dimensional Harmonic oscillator and their descriptions will be also unified.

Keywords:Matrix, Operator, Wave function, Harmonic oscillator

Introduction

In 1925, Heisenberg, Born and Jordan created matrix mechanics in which they

introduced the matrix to describe the mechanical quantity. The following matrix was

used to describe the position x of a particle in one-dimensional harmonic oscillator

0 1 0 0

1 0 0 01

20 0 0 1

0 0 1 0

s

s

a

-

-

(1)


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The creators of matrix mechanics found their matrices such as (1) in special cases by

guesswork, guided by the correspondence principle. [2]

In 1926, Schrdinger presented what is now known as the Schrdinger equation and

arrived at his theory of wave mechanics. For one-dimensional harmonic oscillator, the

Schrdinger equation is

2 22 2

2

1( )

2 2

dx E

dxmw y y

m- + =h

(2)

The solutions to (2) are (29) and (37)-(38). Let

x x= , p ix= - h (3)

Although Schrdinger had shown that his theory was mathematically equivalent to

matrix mechanics, [3] he had not revealed the direct relations between wave

mechanics and matrix mechanics.

In fact, from (3) and (37)-(38),

22

22

1 1

4 2 21 2

1 324 2 2

2 1 3

1 1

1

2

1 1 2

2

1 ( , 0)

2

x

x

s s s

x xe

x x e

sx s

a

a

y p a y a

y p a y y aa

y y ya

- -

- -

- +

= =

= = + -

=

(4)

According to the inverse law of matrix multiplication, [1] (4) can be represented as

[ ] [ ]1 2 1 1 2 1 s s s sx y y y y y y y y - -=

0 1 0 0

1 0 0 01

20 0 0 1

0 0 1 0

s

s

a

-

-

(5)

The square matrix in (5) is just (1).


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In 1930, Paul Dirac published his book Principles of Quantum Mechanics. In that

book, the measurable quantities are associated with operators acting on the Hilbert

space of vectors that describe the state of a physical system. [4] The ladder operators

are defined as

21 ( )2

ia x pa

a= +

h (6)

21 ( )2

ia x pa

a

+ = -h

(7)

Combining with (17-20),

1

2 ( )H a aw += +h , [ , ]H a aw= -h , [ , ]H a aw+ +=h , [ , ] 1a a+ = (8)

Let H be an eigenvalue of H and |H an eigenket belonging to it. Dirac regarded

| |H a a H+ as the square of the length of the ket |a H , and if |H is normalized,

then

12 | | 0H H a a Hw w +- = h h (9)

the case of equality occurring only if | 0a H = . As a result, Dirac obtained the

lowest eigenvalue and an eigenket belonging to it. The commutation property yields

| ( ) |Ha H H a Hw = - h , | ( ) |Ha H H a Hw+ + = + h

This means

| |a H a H w = - h , | |a H c H w+ = + h (10)

1( )

2H n w= +h ( 0,1, ,n s= ) (11)

From (10),

* | ( | ) ( | ) |H a a H a H H aw w+ + + = = - = -h h (12)

2 | | | |a H a a H n+= = , 2

| | | | 1c H aa H n+= = +


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Thus

| |a H n H w = - h (13)

| 1 |a H n H w+ = + + h

(14)

It is Diracs great contribution to associate the measurable quantities with operators.

However, his space of state vectors is so abstract that it is difficult to understand it. In

this study, we offer the following new concepts and approaches to improve Diracs

system.

I. A set of orthonormal wave functions describe the quantum state of a physical

system. A quantum state corresponds to a specific wave function.

Similarly to (5), the operator, wave functions and square matrix can be represented in

the same representation. Now that it is easy to associate the square matrix in matrix

mechanics with the operator and wave functions in wave mechanics, why do we have

to introduce state vector to describe quantum states? We do not need the state vector

which describes quantum states. Dirac had given Bra-ket notation too many meanings.

For example,

| |A A + = (15)

In our system, Bra-ket notation is only used for the inner product and has formal

meaning, given by two definitions in reference 1.

II. The operators are simply some derivative and function symbols. When an operator

acts on a function, it holds the numerical meaning.

In Dirac system, | |A A is regard as an operator and the operator can operate on not

only ket vector but also bra vector such as (12).

In our system, an operator can operator only on the function. Those concepts such as

| |A A and |B A are not defined.


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III. The matrix will be introduced so that some concepts become clearer.

In Diracs system, the notation such as |A + and A+ was also given the meaning of

conjugate transpose belonging to the matrix.

In our system, only the matrices have the meaning of conjugate transpose. According

to the forward and inverse law of matrix multiplication,

[ ] [ ]1 1 2 1 2 1 22 1 2

A a b a cA

b dA c d

y y yy y y y

y y y

= + =

= + (16)

Thus, the differential equations can be associated with the matrix. The property of the

operator A such as Hermitain property reflects on the square matrix in (16). We can

determine the matrix elements in the square matrix when the wave functions are

known; on the other hand, when the matrix elements in the square matrix are known,

we can also determine the differential equations. In fact, if 1y and 2y are

orthonormal, then according to two definitions in reference 1, we can get from (16)

1 1 1 2 11 2

22 1 2 2

| | | | | || |

A A a cA A

b dA A

y y y y y y y

yy y y y

= =

This is just so-called matrix representation of the operator A in representation

theory in Diracs system.

IV. The following postulates of quantum mechanics become naturally the starting

points in our system.

1). Born's probability interpretation.

2).There are the eigenequations including the Schrdinger equation.

3).The principle of superposition states.

4). In quantum mechanics, the operators that describe mechanical quantities are

Hermitain operators.


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One-dimensional nondegenerate harmonic oscillator

The Hamiltonian operator of one-dimensional nondegenerate harmonic oscillator is

written as

2 1 2 2 2 2

pH xmw

m= + (17)

Combining with (3), we can show that

[ , ]x p i=h (18)

[ , ]x H i p=h

(19)

2 [ , ]p H i xw= -h (20)

The Schrdinger equation isH Ey y=

The energy levels are discrete in quantum mechanics. Thus, let 1 2, , , sE E E be the

eigenvalues of H and 1 2, , , sy y y the orthonormalized eigenfunctions belongingto it in terms of Born's probability interpretation, then

1

21 2 1 2

0 0

0 0 [ ] [ ]

0 0

s s

s

E

EH

E

y y y y y y

=

(21)

where 1 2, , , sE E E and 1 2, , , sy y y are all unknown.

Because x is a Hermitian operator, according to the theorem in reference 1 which

based on the above four postulates of quantum mechanics, it is assumed that

* *11 21 1

*21 22 2

1 2 1 2

1 2

[ ] [ ]

s

s

s s

s s ss

X X X

X X Xx

X X X

y y y y y y

=

(22)

where 11 22, , , ssX X X are real number and these matrix elements in the square


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matrix are all unknown. From (19), (21) and (22),

1 2 1 2[ ] [ ]s sp y y y y y y =

* *

1 2 21 1 1*2 1 21 2 2

1 1 2 2

0 ( ) ( )

( ) 0 ( )

( ) ( ) 0

s s

s s

s s s s

E E X E E X

E E X E E Xi

E E X E E X

m

- - - -

- -

h (23)

From (20)(21) and (23),

1 2 1 2[ ] [ ]s sx y y y y y y =

2 * 2 *

2 1 21 1 12 2 *

2 1 21 2 22 2

2 21 1 2 2

0 ( ) ( )( ) 0 ( )1

( ) ( ) 0

s s

s s

s s s s

E E X E E XE E X E E X

E E X E E X

w

- - - -

- -

h (24)

Comparison of (22) with (24) yields

2 * 2 * 2 *2 1 21 3 1 31 1 1

2 2 * 2 *2 1 21 3 2 32 2 2

2 2 2 *

3 1 31 3 2 32 3 3

2 2 21 1 2 2 3 3

0 ( ) ( ) ( )

( ) 0 ( ) ( )

( ) ( ) 0 ( )

( ) ( ) ( ) 0

s s

s s

s s

s s s s s s

E E X E E X E E X

E E X E E X E E X

E E X E E X E E X

E E X E E X E E X

- - -

- - -

- - - - - -

* * *11 21 31 1

* *21 22 32 2

2 2 *31 32 33 3

1 2 3

s

s

s

s s s ss

X X X X

X X X X

X X X X

X X X X

w

=

h

Therefore 11X = 22X == ssX =0

Because 1 2 3 1, , , , ,s sE E E E E- are real number and it is clear that

sE > 1 3 2 1sE E E E- > > > > therefore

2 1E E w- =h

3 2 31, 0E E Xw- = =h


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1 1 2, 0s s s ssE E X Xw- -- = = = =h

Thus, (22) and (24) become

[ ] [ ]1 2 3 2 1 1 2 3 2 1 s s s s s sx y y y y y y y y y y y y - - - -=

*21

*21 32

32

*1 2

*1 2 1

1

0 0 0 0 0

0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00 0 0 0 0

s s

s s ss

ss

X

X X

X

X

X X

X

- -

- - -

-

(25)

[ ] [ ]1 2 3 2 1 1 2 3 2 1 s s s s s sp y y y y y y y y y y y y - - - -=

*21

*21 32

32

*1 2

*1 2 1

1

0 0 0 0 0

0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0

s s

s s ss

ss

X

X X

X

i

X

X X

X

mw

- -

- - -

-

-

-

- -

(26)

Thus, from (18), (25) and (26),

2 2 221 32 12 2 2

1 1 1| | ,| | , ,| |

2 2sss

X X Xa a a

-

-= = = (

mwa=

h) (27)

21 2

1| | 0

2ss s s sX y y y

a-- = = (28)

From (17), (21) and (25)-(27),

1

1

2E w=h , 2

3

2E w=h , ,

1( )

2sE s w= -h (29)

From (6), (7) and (25)-(26),

[ ] [ ]1 2 1 1 2 1 s s s sa y y y y y y y y - -=


9/12

*21

*1

0 0 0

0 0 0 0

2

0 0 0

0 0 0 0ss

X

X

a

-

(30)

[ ] [ ]1 2 1 1 2 1 s s s sa y y y y y y y y +

- -=

21

1

0 0 0 0

0 0 0

2

0 0 0 0

0 0 0ss

X

X

a

-

(31)

If we take positive real solutions from (27), then we have

21 32 1

1 2 1, , ,

2 2 2ss

sX X X

a a a-

-= = = (32)

Thus, (30) and (31) become

[ ] [ ]1 2 1 1 2 1 s s s sa y y y y y y y y - -=

0 1 0 0

0 0 0 0

0 0 0 1

0 0 0 0

s

-

(33)

[ ] [ ]1 2 1 1 2 1 s s s sa y y y y y y y y +

- -=

0 0 0 0

1 0 0 0

0 0 0 0

0 0 1 0s

-

(34)

Thus:

1 2 1 1 0, , , 1s sa a a sy y y y y -= = = - (35)


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1 2 2 3 1 , 2 , , 1 , 0s s sa a a s ay y y y y y y + + + +

-= = = - = (36)

From (35),

221 1

4 2 21

xe

a

y p a- -

= (37)

It can be followed by the other wave functions in terms of (36)

2 22 21 3 1 1 1 1

4 2 2 2 2 4 2 22 12 , , 2 [( 1)!] ( )

sx x

s sx e s H x ea a

y p a y a p a -

- - - - - -

-= = - (38)

The functions 1( )sH xa- are the Hermite polynomials.

2 21

11 1( ) ( 1) ( )

ss z z

s s

dH z e e z x

dz

a-

- -- -

= - =

when s , 0sy =

Discussion

I. In Diracs system, he tried to represent | |a H a H w = - h in (10) as

[ ] | / 2 | 3 / 2 | (2 1) / 2 | (2 1) / 2a s sw w w w - + h h h h

[ ]| / 2 | 3 / 2 | (2 1) / 2 | (2 1) / 2s sw w w w = - + h h h h

*1

*

0 0 0

0 0 0 0

0 0 0

0 0 0 0s

a

a

(39)

Furthermore, from (12),

1

0 0 0 0/ 2 | / 2 |

0 0 03 / 2 | 3 / 2 |

0 0 0 0(2 1) / 2 | (2 1) / 2 |

0 0 0(2 1) / 2 | (2 1) / 2 |s

a

a

s s

as s

w w

w w

w w

w w

+

=

- - + +

h h

h h

h h

h h

(40)


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In fact, (39) corresponds to (30). (40) depends on | ( | )H a a H+ + = and the

definition of the operator operating on bra vector |H a+ .

In our system, (31) is derived from (22).II.Dirac introduced Schrdingers representation with x diagonal and obtained the

representatives of the stationary states. He regarded | 0x as the ground state wave

function. Furthermore, with the help of the following expression

1 1| |n r nr

x x p i x xx

= -

h ,

he obtained the ground state wave function.

In fact, according to (13) and (35), the state vector | 0 corresponds to the wave

function 1y . We obtained (37) without invoking assistance from the Dirac concepts

of the state vectors. Similarly, for the angular momentum, the state vector | jm

corresponds to the wave function jmY . [1]

1

1

( 1)( )

( )( 1)

jm jm

z jm jm

jm jm

L Y j m j m Y

L Y m Y

L Y j m j m Y

+ +

- -

= + + -

=

= + - +

h

h

h

( ,1 , , ; 0,1, )m j j j j= - - =

We can get the solutions of these differential equations [1] which are also the

solutions of the following differential equations.

2

2 2 22 21 1 [ (sin ) ] ( , ) ( , ) ( , )sin sin

( , ) ( , ) ( , )z z

Y L Y Y

i Y L Y L Y

q q j q j l q j q q q q j

q j q j q j j

- + = =

- = =

h h

h

For example

00 0000

0000

( cot ) 0

0

i

z

Y Ye i L Y

Yi L Y

j qq j

j

+

+ = =

- = =

h

h


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The solution of the differential equations is 00 1/ 4Y p= .

The concepts, processes and the results in our system do not depend on those in

Diracs system. Therefore, we obtained the results of matrix mechanics and wave

mechanics without Diarcs abstract concepts and definitions and unified the

description of them.

Acknowledgments: thank Ph. D Dinghan Chen for some advices.

Reference

1. Yongqin Wang, Lifeng Kang, Unified Description of Matrix Mechanics and Wave

Mechanics on Hydrogen Atom, http://arxiv.org/abs/1201.0136

2B.L.van der Waerden, Sources of Quantum Mechanics, pp. 297-306. Dover

Publications, Inc., New York (1968)

3. Schrdinger, E., ber das Verhltnis der Heisenberg Born Jordanischen

Quantenmechanik zu der meinen, Annalen der Physik. Leipzig 79(1926) 734

4. P.A.M. Dirac, the Principle of Quantum Mechanics (Fourth Edition), pp. 136-139.

Oxford at the Clarendon Press, London (1958)

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Unified Representation of Quantum Mechanics on.pdf