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Redalyc.Harmonic Oscillator Position Eigenstates Via Application of An Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Soto-Eguibar, Francisco; Moya-Cessa, Héctor Manuel Harmonic oscillator position eigenstates via application of an operator on the vacuum Revista Mexicana de Física, vol. 59, núm. 2, julio-diciembre, 2013, pp. 122-127 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57048159005 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative EDUCATION Revista Mexicana de F´ısica E 59 (2013) 122–127 JULY–DECEMBER 2013 Harmonic oscillator position eigenstates via application of an operator on the vacuum Francisco Soto-Eguibar and Hector´ Manuel Moya-Cessa Instituto Nacional de Astrof´ısica, Optica´ y Electronica,´ Luis Enrique Erro 1, Santa Mar´ıa Tonantzintla, San Andres´ Cholula, Puebla, 72840 Mexico.´ Received 21 August 2013; accepted 17 October 2013 Harmonic oscillator squeezed states are states of minimum uncertainty, but unlike coherent states, in which the uncertainty in position and momentum are equal, squeezed states have the uncertainty reduced, either in position or in momentum, while still minimizing the uncertainty principle. It seems that this property of squeezed states would allow to obtain the position eigenstates as a limiting case, by doing null the uncertainty in position and infinite in momentum. However, there are two equivalent ways to define squeezed states, that lead to different expressions for the limiting states. In this work, we analyze both definitions and show the advantages and disadvantages of using them in order to find position eigenstates. With this in mind, but leaving aside the definitions of squeezed states, we find an operator that applied to the vacuum gives position eigenstates. We also analyze some properties of the squeezed states, based on the new expressions obtained for the eigenstates of the position. Keywords: Position eigenstates; harmonic oscillator; squeezed states; minimum uncertainty states; squeeze operator. PACS: 03.65.-w 1. Introduction duce the uncertainties either of the position or momentum, while still keeping the uncertainty principle to its minimum. Quantum states of the harmonic oscillator are very important Because of this, they belong to a special class of states named in the study of the quantum theory of the electromagnetic minimum uncertainty states. Once produced, for instance as field because of the fact that they may be used to describe electromagnetic fields in cavities, they may be monitored via a quantized field [1–3], each harmonic oscillator represent- two level atoms in order to check, or measure, that such states ing a mode of such electromagnetic field. In fact, this was have been indeed generated [10, 11]. the concept that Dirac used to build the first quantum theory Based on the above properties, we can think about eigen- of the electromagnetic field [2]. The easiest to understand states of position as limiting cases of squeezed states. As and to manipulate, and the most natural states of the quantum squeezed states are minimum uncertainty states, we can re- harmonic oscillator are number states jni. Number states are duce to zero the uncertainty in the position, while the uncer- eigenstates of the harmonic oscillator Hamiltonian and, of tainty in the momentum goes to infinity, so that we keep the course, are also eigenstates of the number operator n^ =a ^ya^, uncertainty principle to its minimum. Of course, there is also where a^y and a^ are the well known creation and annihilation the option to reduce to zero the uncertainty in the momen- operators, respectively. However, for any n, no matter how tum, while the position gets completely undefined, obtain- big, the mean field is zero; i.e., hnjE^xjni = 0, and we know ing that way the possibility to define momentum eigenstates. that a classical field changes sinusoidally in time in each point In Secs. 1 and 2, we analyze the possibility of define the of space; thus, these states can not be associated with classi- position eigenstates as the limit of extreme squeezing of the cal fields [3, 4]. squeezed states. In what follows, we will use a unit system such that ~ = m = ! = 1. In the first years of the sixties of the past century, Glauber [5] and Sudarshan [6] introduced the coherent states, There are two equivalent forms to define the squeezed and it has been shown that these states are the most clas- states. In the first one, introduced by Yuen [12], squeezed sical ones. Coherent states are denoted as j®i, and one states are obtained from the vacuum as way to define them is as eigenstates of the annihilation j®; ri = S^(r)D^(®)j0i = S^(r)j®i; (1) operator; that is, a^j®i = ®j®i. An equivalent defini- tion is obtained¡ applying the¢ Glauber displacement operator y ¤ where h³ ´ i D^(®) = exp ®a^ ¡ ® a^ to the vacuum: j®i = D^(®)j0i; 2 S^(r) = exp a^2 ¡ a^y r=2 (2) we see then coherent states as vacuum displaced states. Co- herent states also have the very important property that they is the so-called squeeze operator. In this view, squeezed states minimize the uncertainty relation for the two orthogonal field are created displacing the vacuum, and after, squeezing it. quadratures with equal uncertainties in each quadrature [3,4]. Note that when the squeeze parameter r is zero, the squeezed Since then, other states have been introduced. In partic- states reduce to the coherent states. In this work, we will con- ular, squeezed states [3, 4, 7–9] have attracted a great deal of sider only real squeeze parameters, as that is enough for our attention over the years because their properties allow to re- intentions. HARMONIC OSCILLATOR POSITION EIGENSTATES VIA APPLICATION OF AN OPERATOR ON THE VACUUM 123 In the definition introduced by Caves [13], the vacuum is Proceeding in exactly the same way for the quadrature oper- squeezed and the resulting state is then displaced; that means, ator Y^ , we obtain q that in this approach squeezed states are given as er ¢ Y ´ h®; rjY^ 2j®; ri ¡ h®; rjY^ j®; ri2 = : (15) 2 j®0; r0i = D^ (®0) S^ (r0) j0i: (3) As we already said, we can then think in the position eigen- Both definitions of the squeezed states agree when the states and in the momentum eigenstates as limit cases of squeeze factor is the same, r0 = r, and when the modified squeezed states. Indeed, when the squeeze parameter r goes amplitude ®0 of the Caves approach is given by to infinity, the uncertainty in the position goes to zero, and the momentum is completely undetermined. Of course, when the 0 ¤ ® = ¹® ¡ º® ; (4) squeeze parameter goes to minus infinity, we have the inverse situation, and we can think in define that way the momentum being eigenstates. In the two following sections, we use the Yuen ¹ = cosh r (5) and the Caves definitions of the squeezed states to test this and hypothesis. º = sinh r: (6) To analyze the uncertainties in the position and in the 2. A first attempt a` la Yuen momentum of the squeezed states, we introduce, following Loudon and Knight [7], the quadrature operators From Eq. (14) above, we can see that in the limit r ! 1 the uncertainty for position vanishes and so a position eigenstate a^ +a ^y x^ X^ = = p (7) should be obtained (from now on, we consider ® real), 2 2 x lim jp ; ri ! jxip: (16) and r!1 2 a^ ¡ a^y p^ Y^ = = p (8) We have written a sub index p in the position eigenstate in 2i 2 order to emphasis that fact. Following the Yuen definition where x^ is the position operator and p^ the momentum opera- j®; ri = S^(r)D^(®)j0i = S^(r)j®i, so µ ¶ tor. Note that the quadrature operators are essentially the po- x x x jp ; ri = S^(r)D^ p j0i = S^(r)jp i: (17) sition and momentum operators; this definition just provides 2 2 2 us with two operators that have the same dimensions. We now write the squeeze operator as [16] In order to show that really the squeezed states are min- 1 º y2 1 º 2 imum uncertainty states, we need to calculate the expected ^ ¡ 2¹ a^ 2¹ a^ S(r) = p e y e ; (18) values in the squeezed state (1) of the quadrature operators ¹ ¹a^ a^ (7) and (8), and its squares. Using (7) and (1), we get where, as we already said, ¹ = cosh r and º = sinh r. So, 2 y x 1 ¡ º a^y 1 º a^2 x 1 y a^ +a ^ jp ; ri = e 2¹ e 2¹ jp i: (19) h®; rjX^j®; ri = h®jS^ (r) S^(r)j®i: (9) p a^ya^ 2 2 2 ¹ ¹ 2 The action of the squeeze operator on the creation and Now, we develop the first operator (from right to left) in annihilation operators is obtained using the Hadamard’s power series, we use the definition of the coherent states, lemma [14, 15], a^j®i = ®j®i, and¡ the action of the number¢ operator over the number states a^ya^jni =n ^jni = njni , to obtain ^y ^ y ^y y ^ y S (r)^aS(r) = ¹a^ ¡ ºa^ ; S (r)^a S(r) = ¹a^ ¡ ºa;^ (10) µ ¶a^ya^ 1 µ ¶n 2 X x 1 ¡ º a^y 1 x 1 jp ; ri = p e 2¹ p p jni such that ¹ ¹ y 2 n=0 2 n! ^y a^ +a ^ ^ ¡r ^ S (r) S(r) = e X: (11) 1 µ ¶n µ ¶n 2 2 X 1 ¡ º a^y x 1 1 y ¤ = p e 2¹ p p jni: (20) Therefore, as a^j®i = ®j®i and h®ja^ = h®j® , it is easy to ¹ 2 n! ¹ see that n=0 ¤ 1 1 ¡r ® + ® r ! 1 = ! 0 h®; rjX^j®; ri = e ; (12) As , ¹ cosh r , which means that the only term 2 that survives from the sum is n = 0, and then and that º y2 ¡ 2¹ a^ 2 jxip / e j0i (21) 1 + 2 j®j2 + ®2 + ®¤ h®; rjX^ 2j®; ri = e¡2r : (13) that would give an approximation for how to obtain a po- 4 sition eigenstate from the vacuum.
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