#### Question Details

(solution) Explain, in your own words (without going into mathematical

Explain, in your own words (without going into mathematical details, although you may use the odd equation for illustrative purposes), the main features of the Heisenberg-Born-Jordan atom, as described by their new theory of Quantum Mechanics, using quotes from the papers of Born and Jordan and the analysis of that paper by Fedak and Prentis. The explanation should include: what are the new variables describing the states of the atom? What are their fundamental properties? How do they link to observables? The two papers are attached.

Also add references

1925 M. Born, Z. Phys. 34, 858 On Quantum Mechanics M. Born

Received 1925

?? ? ??

Translation into English: Sources of Quantum Mechanics,

Ed. by B. L. van der Waerden, North Holland, Amsterdam (1967) 277.

?? ? ?? The recently published theoretical approach of Heisenberg is here developed

into a systematic theory of quantum mechanics (in the first place for systems having one degree of freedom) with the aid of mathematical matrix methods. After

a brief survey of the latter, the mechanical equations of motion are derived from

a variational principle and it is shown that using Heisenberg?s quantum condition,

the principle of energy conservation and Bohr?s frequency condition follow from the

mechanical equations. Using the anharmonic oscillator as example, the question of

uniqueness of the solution and of the significance of the phases of the partial vibrations is raised. The paper concludes with an attempt to incorporate electromagnetic

field laws into the new theory. Introduction

The theoretical approach of Heisenberg 1 recently published in this Journal,

which aimed at setting up a new kinematical and mechanical formalism in

conformity with the basic requirements of quantum theory, appears to us

of considerable potential significance. It represents an attempt to render

1 W.Heisenberg, Zs. f. Phys. 33 (1925) 879. 1 justice to the new facts by selling up a new and really suitable conceptual

system instead of adapting the customary conceptions in a more or less artificial and forced manner. The physical reasoning which led Heisenberg

to this development has been so clearly described by him that any supplementary remarks appear superfluous. But, as he himself indicates, in its

formal, mathematical aspects his approach is but in its initial stages. His

hypotheses have been applied only to simple examples without being fully

carried through to a generalized theory. Having been in an advantageous

position to familiarize ourselves with his ideas throughout their formative

stages, we now strive (since his investigations have been concluded) to clarify the mathematically formal content of his approach and present some of

our results here. These indicate that it is in fact possible, starting with

the basic premises given by Heisenberg, to build up a closed mathematical theory of quantum mechanics which displays strikingly close analogies

with classical mechanics, but at the same time preserves the characteristic

features of quantum phenomena.

In this we at first confine ourselves, like Heisenberg, to systems having one degree of freedom and assume these to be ? from a classical

standpoint ? periodic. We shall in the continuation of this publication concern ourselves with the generalization of the mathematical theory to systems having ah arbitrary number of degrees of freedom, as also to aperiodic

motion. A noteworthy generalization of Heisenberg?s approach lies in our

confining ourselves neither to treatment of nonrelativistic mechanics nor to

calculations involving Cartesian systems of coordinates. The only restriction

which we impose upon the choice of coordinates is to base our considerations

upon libration coordinates, which in classical theory are periodic functions

of time. Admittedly, in some instances it might be more reasonable to

employ other coordinates: for example, in the case of a rotating body to

introduce the angle of rotation ?, which becomes a linear function of time.

Heisenberg also proceeded thus in his treatment of the rotator; however, it

remains undecided whether the approach applied there can be justified from

the standpoint of a consistent quantum mechanics.

The mathematical basis of Heisenberg?s treatment is the law of multiplication of quantum?theoretical quantities, which he derived from an

ingenious consideration of correspondence arguments. The development of

his formalism, which we give here, is based upon the fact that this rule of

multiplication is none other than the well?known mathematical rule of matrix multiplication. The infinite square array (with discrete or continuous

indices) which appears at the start of the next section, termed a matrix, is

a representation of a physical quantity which is given in classical theory as

2 a function of time. The mathematical method of treatment inherent in the

new quantum mechanics is thereby characterized through the employment

of matrix analysis in place of the usual number analysis.

Using this method, we have attempted to tackle some of the simplest

problems in mechanics and electrodynamics. A variational principle, derived from correspondence considerations, yields equations of motion for the

most general Hamilton function which are in closest analogy with the classical canonical equations. The quantum condition conjoined with one of the

relations which proceed from the equations of motion permits a simple matrix notation. With the aid of this, one can prove the general validity of the

law of conservation of energy and the Bohr frequency relation in the sense

conjectured by Heisenberg: this proof could not be carried through in its

entirety by him even for the simple examples which he considered. We shall

later return in more detail to one of these examples in order to derive a basis

for consideration of the part played by the phases of the partial vibrations in

the new theory. We show finally that the basic laws of the electromagnetic

field in a vacuum can readily be incorporated and we furnish substantiation

for the assumption made by Heisenberg that the squares of the absolute

values of the elements in a matrix representing the electrical moment of an

atom provide a measure for the transition probabilities. Chapter 1. Matrix Calculation

1. Elementary operations. Functions

We consider square infinite matrices, 2 which we shall denote by heavy

type to distinguish them from ordinary quantities which will throughout be

in light type, a(00) a(01) a(02) · · · a(10) a(11) a(12) · · · a = (a(nm)) = a(20) a(21) a(22) · · · .

···

···

···

2 Further details of matrix algebra can be found, e.g., in M. Bocher, Einf¨

uhrung in die

h¨

ohere Algebra (translated from the English by Hans Beck; Teubner, Leipzig, 1910) § 22?

25; also in R. Courant and D. Hilbert, Methoden der mathematischen Physik 1 (Springer,

Berlin, 1924) Chapter I. 3 Equality of two matrices is defined as equality of corresponding components:

a=b means a(nm) = b(nm). (1) Matrix addition is defined as addition of corresponding components:

a=b+c means a(nm) = b(nm) + c(nm). (2) Matrix multiplication is defined by the rule ?rows times columns?, familiar

from the theory of determinants:

a = bc means a(nm) = ?

X b(nk)c(km). (3) k=0 Powers are defined by repeated multiplication. The associative rule applies

to multiplication and the distributive rule to combined addition and multiplication:

(ab)c = a(bc);

(4)

a(b + c) = ab + ac. (5) However, the commutative rule does not hold for multiplication: it is not in

general correct to set ab = ba. If a and b do satisfy this relation, they are

said to commute.

The unit matrix defined by

?nm = 0 for n 6= m,

(6)

1 = (?nm ),

?nm = 1

has the property

a1 = 1a = a. (6a) The reciprocal matrix to a, namely a?1 , is defined by3

a?1 a = aa?1 = 1 (7) As mean value of a matrix a we denote that matrix whose diagonal elements

are the same as those of a whereas all other elements vanish:

?

a = (?nm a(nm)). (8) As is known, a?1 is uniquely defined by (7) for finite square matrices when the determinant A of the matrix a is non?zero. If A = 0 there is no matrix to a.

3 4 The sum of these diagonal elements will be termed the diagonal sum of the

matrix a and written as D(a), viz.

X

D(a) =

a(nm).

(9)

n From (3) it is easy to prove that if the diagonal sum of a product

y = x1 x2 · · · xm be finite, then it is unchanged by cyclic rearrangement of

the factors:

D(x1 x2 · · · xm ) = D(xr xr+1 · · · xm x1 x2 · · · xr?1 ). (10) Clearly, it suffices to establish the validity of this rule for two factors.

If the elements of the matrices a and b are functions of a parameter t,

then

X

d X

a(nk)b(km) =

{a(nk)b(km)

?

+ a(nk)b(km)},

dt n

k or from the definition (3): d

?

(ab) = ab

? + ab.

dt (11) Repeated application of (11)

d

(x1 x2 · · · xn ) = x? 1 x2 · · · xn + x1 x? 2 · · · xn + · · · + x1 x2 · · · x? n .

dt (110 ) From the definitions (2) and (3) we can define functions of matrices. To begin

with, we consider as the most general function of this type, f (x1 , x2 · · · xm ),

one which can formally be represented as a sum of a finite or infinite number

of products of powers of the arguments xk ; weighted by numerical coefficients. Through the equations

f1 (y1 , · · · yn ; x1 , · · · xn ) = 0,

............................

fn (y1 , · · · yn ; x1 , · · · xn ) = 0 (12) we can then also define functions yl (x1 , . . . xn ); namely, in order to obtain

functions yl ; having the above form and satisfying equation (12), the yl (

need only be set in form of a series in increasing power products of the xk and

the coefficients determined through substitution in (12). It can be seen that

one will always derive as many equations as there are unknowns. Naturally,

the number of equations and unknowns exceeds that which would ensue

5 from applying the method of undetermined coefficients in the normal type of

analysis incorporating commutative multiplication. In each of the equations

(12), upon substituting the series for the yl ; and gathering together like

terms one obtains not only a sum term C 0 x1 x2 but also a term C 00 x2 x1 and

thereby has to bring both C 0 and C? to vanish (e.g., not only C 0 + C?). This

is, however, made possible by the fact that in the expansion of each of the

yl , two terms x1 x2 and x2 x1 appear, with two available coefficients. 2. Symbolic differentiation

At this stage we have to examine in detail the process of differentiation

of a matrix function, which will later be employed frequently in calculation.

One should at the outset note that only in a few respects does this process

display similarity to that of differentiation in ordinary analysis. For example,

the rules for differentiation of a product or of a function of a function here

no longer apply in general. Only if all the matrices which occur commute

with one another can one apply all the rules of normal analysis to this

differentiation.

Suppose

s

Y

xlm = xl1 xl2 . . . xls .

(13)

y=

m=1 We define s

s

X

Y

?y

=

?lr k

?xk

r=1 m=r+1 xlm m=r?1

Y xlm , m=1 ?jk = 0 for

?kk = 1. j 6= k, (14) This rule may be expressed as follows: In the given product, one regards all factors as written out individually (e.g., not as x31 x22 , but as

x1 x1 x1 x2 x2 ); one then picks out any factor xk and builds the product

of all the factors which follow this and which precede (in this sequence).

The sum of all such expressions is the differential coefficient of the product

with respect to this xk .

The procedure may be illustrated by some examples:

y = xn , dy

= nxn?1

dx y = xn1 xm

2 , ?y

n?2 m

n?1

= x1n?1 xm

x2 x1 + · · · + xm

,

2 + x1

2 x1

?x1 y = x21 x2 x1 x3 , ?y

= x1 x2 x1 x3 + x2 x1 x3 x1 + x3 x21 x2 .

?x1

6 If we further stipulate that

?y1

?y2

?(y1 + y2 )

=

+

,

?xk

?xk

?xk (15) then the derivative ?y/?x is defined for the most general analytical functions

y.

With the above definitions, together with that of the diagonal sum (9),

there follows the relation

?D(y)

?y

(mn),

=

?xk (nm)

?xk (16) on the right?hand side of which stands the mn?component of the matrix

?y/?xk . This relation can also be used to define the derivative ?y/?xk . In

order to prove (16), it obviously suffices to consider a function y having the

form (13). From (14) and (3) it follows that

s

s

X

X Y

?y

?lr k

=

?xk (mn)

?

r=1 xlp (?p ?p+1 ) p=r+1 ?r+1 = m, r?1

Y xlp (?p ?p+1 ); (17) p=1 ?s+1 = ?1 , ?r = n. On the other hand, from (3) and (9) ensues

s

s

X

X r?1

Y

Y

?D(y)

?lr k

xlp (?p ?p+1 )

=

?xk (mn)

?

r=1 p=1 ?1 = ?s+1 , xlp (?p ?p+1 ); (170 ) p=r+1 ?r = n, ?r+1 = m. Comparison of (17) with (17?) yields (16).

We here pick out a fact which will later assume importance and which

can be deduced from the definition (14): the partial derivatives of a product

are invariant with respect to cyclic rearrangement of the factors. Because of

(16) this can also be inferred from (10).

To conclude this introductory section, some additional description is

devoted to functions g(pq) of the variables. For

y = ps q r (18) it follows from (14) that

s?1 r?1 ?y X s?1?l r l

p

q p,

=

?p ?y X r?1?j s j

q

p q .

=

?q

j=1 l?1 7 (180 ) The most general function g(pq) to be considered is to be represented

in accordance with § 1 by a linear aggregate of terms

k

Y z= (psj qrj ). (19) j=1 With the abbreviation

pl = k

Y (psj qrj ) l?1

Y (psj qrj ), (20) j=1 j=l+1 one can write the derivatives as

k sl ?1

?z = P P

psl ?1?m qrl pl pm ,

?p l=1 m=0

k rP

l ?1

P ?z =

?q l=1 m=0 qrl ?1?m pl psl qm . (21) From these equations we find an important consequence. We consider the

matrices

?z

?z

?z

?z

d1 = q

?

q, d2 = p

?

p.

(22)

?q ?q

?p ?p

From (21) we have

d1 = k

X (qrl Pl psl ? Pl psl qrl ), k

X (psl qrl Pl ? Pl psl qrl ). l=1 d2 = l=1 and thus it follows that d1 + d2 = k

X

l=1 (psl qrl Pl ? Pl psl qrl ). Herein the second member of each term cancels the first member of the

following, and the first and last member of the overall sum also cancel, so

that

(23)

d1 + d2 = 0.

8 Because of its linear character in z, this relation holds not only for expressions z having the form (19), but indeed for arbitrary analytical functions

g(pq).4

In concluding this brief survey of matrix analysis, we establish the following rule: Every matrix equation

F(x1 , x2 , . . . xr ) = 0

remains valid if in all the matrices xj one and the same permutation of all

rows and columns is undertaken. To this end, it suffices to show that for

two matrices a, b which thereby become transposed to a0 , b0 , the following

invariance conditions apply:

a0 + b0 = (a + b)0 , a0 b0 = (ab)0 , wherein the right?hand sides denote those matrices which are formed from

a + b and ab respectively by such an interchange.

We set forth this proof by replacing the procedure of permutation by

that of multiplication with a suitable matrix.5

We write a permutation as

n

0 1 2 3 ...

=

kn

k0 k1 k2 k3 . . .

and to this we assign a permutation matrix,

1 when m = kn

p = (p(nm)), p(nm) =

0 otherwise.

The transposed matrix to p is

p

? = (?

p(nm)),

4 p?(nm) = 1 when n = km

0 otherwise. More generally, for function of r variables, one has

X

?g

?g

xr

?

xr = 0.

?xr

?xr

r 5 The method of proof adopted here possesses the merit of revealing the close connection

of permutations with an important class of more general transformations of matrices. The

validity of the rule in question can however also be established directly on noting that in

the definitions of equality, as also of addition and multiplication of matrices, no use was

made of order relationships between the rows or the columns. 9 On multiplying the two together, one has

X

p?

p=(

p(nk)?

p(km)) = (?nm ) = 1,

k since the two factors p(nk) and p?(km) differ from zero simultaneously only

if k = kn = km , i.e., when n = m. Hence p? is reciprocal to p:

p

? = p?1 .

If now a be any given matrix, then

X

pa = (

p(nk)a(km)) = (a(kn , m))

k is a matrix which arises from the permutation

equivalently

ap?1 = ( X n

kn of the rows of a and a(mk)?

p(km)) = (a(n, km )) k is the matrix arising from permutation of the columns of a. One and the

same permutation applied both to the rows and the columns of a thus yields

the matrix

a0 = pap?1 .

Thence follows directly

a0 + b0 = p(a + b)p?1 = (a + b)0 ,

= (ab)0

a0 b0 = pabp?1

which proves our original contention.

It is thus apparent that from matrix equations one can never determine

any given sequence or order of rank of the matrix elements. Moreover, it

is evident that a much more general rule applies, namely that every matrix

equation is invariant with respect to transformations of the type

a0 = bab?1 ,

where b denotes an arbitrary matrix. We shall sec later that this does not

necessarily always apply to matrix differential equations. 10 Chapter 2. Dynamics

3. The basic laws

The dynamic system is to be described by (lie spatial coordinate q and

the momentum p, these being represented by matrices

q = (q(nm)e2?i?(nm)t , p(p(nm)e2?i?(nm)t ). (24) Here the ?(nm) denote the quantum-theoretical frequencies associated with

transitions between states described by the quantum numbers n and m. The

matrices (24) are to be Hermitian, e.g., on transposition of the matrices,

each element is to go over into its complex conjugate value, a condition

which should apply for all real t. We thus have

q(nm)q(mn) = |q(nm))|2 (25) ?(nm) = ??(mn). (26) and

If q be a Cartesian coordinate, then the expression (25) is a measure of the

probabilities6 of the transitions n ? m.

Further, we shall require that

?(jk) + ?(kl) + ?(lj) = 0. (27) This can be expressed together with (26) in the following manner: there

exist quantities Wn such that

h?(nm) = Wn ? Wm . (28) From this, with equations (2), (3), it follows that a function g(pq) invariably

again takes on the form

g = (g(nm)e2?i?(nm)t ) (29) and the matrix (g(nm)) therein results from identically the same process

applied to the matrices (q(nm)), (p(nm)) as was employed to find g from

q, p. For this reason we can henceforth abandon the representation (24) in

favour of the shorter notation

q = (q(nm)),

6 p = (p(nm)). In this connection see §8. 11 (30) For the time derivative of the matrix g = (g(nm)), recalling to mind (24)

or (29), we obtain the matrix

g? = 2?i(?(nm)g(nm)). (31) If ?(nm) 6= 0 when n 6= m, a condition which we wish to assume, then the

formula g? = 0 denotes that g is a diagonal matrix with g(nm) = ?nm (nn).

A matrix differential equation g? = a is invariant with respect to that

process in which the same permutation is carried out on rows and columns

of all the matrices and also upon the numbers Wn In order to realize this,

consider the diagonal matrix

W = (?nm Wn ).

Then

Wg = ( X ?nk Wn g(km)) = (Wn g(nm)), k gW = ( X g(nk)?km Wk ) = (Wm g(nm)), k i.e., according to (31),

g? = 2?i

2?i

((Wn ? Wm )g(nm)) =

(Wg ? gW).

h

h If now p be a permutation matrix, then the transform of W,

W0 = pWp?1 = (?nk m Wnk )

is the diagonal matrix with the permuted Wn along the diagonal. Thence

one has

2?i

pgp

? ?1 =

(W0 g0 ? g0 W0 ) = g? 0 ,

h

where g0 = pgp?1 and g? 0 denotes the time derivative of g0 constructed in

accordance with the rule (31) with permuted Wn .

The rows and columns of g? thus experience the same permutation as

those of g, and hence our contention is vindicated.

It is to be noted that a corresponding rule does not apply to arbitrary

transformations of the form a0 = bab?1 since for these W0 is no longer a

diagonal matrix. Despite this difficulty, a thorough study of these general

transformations would seem to be called for, since it offers promise of insight 12 into the deeper connections intrinsic to this new theory: we shall later revert

to this point.7

In the case of a Hamilton function having the form

1 2

p + U(q)

2m

we shall assume, as did Heisenberg, that the equations of motion are just of

the same form as in classical theory, so that using the notation of §2 we can

write: 1 p,

=

q? = ?H m

?p

(32) p? = ? ?H = ? ?U . ?q

?q

We now use correspondence considerations to try more generally to elucidate the equations of motion belonging to an arbitrary Hamilton function

H(pq). This is required from the standpoint of relativistic mechanics and in

particular for the treatment of electron motion under the influence of magnetic fields. For in this latter case, the function H cannot in a Cartesian

coordinate system any longer be represented by the sum of two functions of

which one depends only on the momenta and the other on the coordinates.

Classically, equations of motion can be derived from the action principle

H= Zt1 t0 Zt1

Ldt = {pq? ? H(pq)}dt = extremum. (33) t0 If we now envisage the Fourier expansion L substituted in (33) and the time

interval t1 ? t0 taken sufficiently large, we find that only the constant term

of L supplies a contribution to the integral. The form which the action

principle thence acquires suggests the following translation into quantum

mechanics:

P

L(kk) is to be made an extremum:

The diagonal sum D(L) =

k D(L) = D(pq? ? H(pq)) = extremum, (34) namely, by suitable choice of p and q, with ?(nm) kept fixed.

Thus, by setting the derivatives of D(L) with respect to the elements of

p and q equal to zero, one obtains the equations of motion

2?i?(nm)q(nm) =

7 ?D(H)

,

dp(nm) Cf. the continuation of this work, lo lie published forthwith. 13 2?i?(mn)p(mn) ?D(H)

.

?q(mn) From (26), (31) and (16) one observes that these equations of motion

can always be written in canonical form, q? = ?H , ?p

(35) p? = ? ?H .

?q For the quantization condition, Heisenberg employed a relation proposed

by Thomas8 and Kuhn.9 The equation

J= I Z1/?

pdq =

pqdt

?

0 of ?classical? quantum theory can, on introducing the Fourier expansions of

p and q,

?

?

X

X

p? e2?i?? t , q =

q? e2?i?? t ,

p=

? =?? ? =?? be transformed into 1 = 2?i ?

X ? ? =?? ?

(q? p?? ).

?J (36) If therein one lias p = mq,

? one can express the p? in terms of q? and

thence obtain that classical equation which on transformation into a difference equation according to the principle of correspondence yields the formula

of Thomas and Kuhn. Since here the assumption that p = mq? should be

avoided, we are obliged to translate equation (36) directly into a difference

equation.

The following expressions should correspond:

?

X ? =?? ? ?

(q? p?? )

?J with ?

1 X

q(n + ?, n)p(n, n + ? ) ? q(n, n ? ? )p(n ? ?, n));

h ? =??

8

9 W. Thomas, Naturwiss. 13 (1925) 627.

W. Kuhn, Zs. f. Phys. 33 (1925) 408. 14 where in the right-hand expression those q(nm), p(nm) which take on a

negative index are to be set equal to zero. In this way we obtain the quantization condition corresponding to (36) as

X

k (p(nk)q(kn) ? q(nk)p(kn)) = h

.

2?i (37) This is a system of infinitely many equations, namely one for each value

of n.

In particular, for p = mq? this yields

X ?(kn)|q(nk)|2 = k h

,

8? 2 m which, as may easily be verified, agrees with Heisenberg?s form of the quantization condition, or with the Thomas-Kuhn equation. The formula (37)

has to be regarded as the appropriate generalization of this equation.

Incidentally one sees from (37) that the diagonal sum D(pq) necessarily becomes infinite. For otherwise one would have D(pq) ? D(qp) = 0

from whereas (37) leads to D(pq) ? D(qp) = ?. Thus the matrices under

consideration arc never finite.10 4. Consequences. Energy-conservation and frequency laws

The content of the preceding paragraphs furnishes the basic rules of

the new quantum mechanics in their entirety. All other laws of quantum

mechanics, whose general validity is to be verified, must be derivable from

these basic tenets. As instances of such laws to be proved, the law of energy

conservation and the Bohr frequency condition primarily enter into consideration. The law of conservation of energy states that if H be the energy,

? = 0, or that H is a diagonal matrix. The diagonal elements H(nn)

then H

of H are interpreted, according to Heisenberg, as the energies of the various

states of the system and the Bohr frequency condition requires that

h?(nm) = H(nn) ? H(mm),

or

Wn = H(nn) + const.

10

Further, they do not belong to the class of ?bounded? infinite matrices hitherto almost

exclusively investigated by mathematicians. 15 We consider the quantity

d = pq ? qp.

From (11), (35) one finds

?H ?H

?H ?H

d? = pq

? + pq? ? qp

? ? qp? = q

?

q+p

?

p.

?q

?q

?p

?p

Thus from (22), (23) it follows that d? = 0 and d is a diagonal matrix. The

diagonal elements of d are, however, specified just by the quantum condition

(27). Summarizing, we obtain the equation

pq ? qp = h

1,

2?i (38) on introducing the unit matrix 1 defined by (6). We term the equation (38)

the ?stronger quantum condition? and base all further conclusions upon it.

From the form of this equation, we deduce the following: If an equation

(A) be derived from (38), then (A) remains valid if p be replaced by q and

simultaneously h by ?h. For this reason one need for instance derive only

one of the following two equations from (38), which can readily be performed

by induction

h n?1

p

pn q = qpn + n

,

(39)

2?i

h n?1

.

(390 )

qn p = pqn ? n

q

2?i

We shall now prove the energy-conservation and frequency laws, as expressed above, in the first instance for the case

H = H1 (p) + H2 (q).

From the statements of §1, it follows that we may formally replace H1 (p)

and bf H2 (q) by power expansions

X

X

as ps , H 2 =

bs qs .

H1 =

s s Formulae (39) and (39?) indicate that

h ?H ,

Hq ? qH = 2?i

?p h ?H . Hp ? pH = ? 2?i

?p

16 (40) Comparison with the equations of motion (35) yields q? = 2?i (Hq ? qH). h p? = 2?i (Hp ? pH). h H for brevity, one has

Denoting the matrix Hg ? gH by g H H H = b + a ; ab a b from which generally for g = g(pq) one may conclude that 2?i H 2?i

g? =

(Hg ? gH). =...

**Solution details:**

Answered

QUALITY

Approved

ANSWER RATING

This question was answered on: * Jan 30, 2021 *

* * Solution~0001001348.zip (25.37 KB)

This attachment is locked

We have a ready expert answer for this paper which you can use for in-depth understanding, research editing or paraphrasing. You can buy it or order for a fresh, original and plagiarism-free solution (Deadline assured. Flexible pricing. TurnItIn Report provided)

##### Pay using PayPal (No PayPal account Required) or your credit card . All your purchases are securely protected by .

#### About this Question

STATUSAnswered

QUALITYApproved

DATE ANSWEREDJan 30, 2021

EXPERTTutor

ANSWER RATING

#### GET INSTANT HELP/h4>

We have top-notch tutors who can do your essay/homework for you at a reasonable cost and then you can simply use that essay as a template to build your own arguments.

You can also use these solutions:

- As a reference for in-depth understanding of the subject.
- As a source of ideas / reasoning for your own research (if properly referenced)
- For editing and paraphrasing (check your institution's definition of plagiarism and recommended paraphrase).

#### NEW ASSIGNMENT HELP?

### Order New Solution. Quick Turnaround

Click on the button below in order to Order for a New, Original and High-Quality Essay Solutions.
New orders are original solutions *and precise to your writing instruction requirements. Place a New Order using the button below.*

WE GUARANTEE, THAT YOUR PAPER WILL BE WRITTEN FROM SCRATCH AND WITHIN A DEADLINE.