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Reply to “Comment on: ‘Questions concerning the generalized Hartman effect’ [Phys. Lett. A 375 (2011) 3259]”
Physics Letters A 376 (2012) 1403–1404
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Comment
Reply to “Comment on: ‘Questions concerning the generalized Hartman effect’ [Phys. Lett. A 375 (2011) 3259]” Shoju Kudaka a , Shuichi Matsumoto b,∗ a b
Department of Physics, University of the Ryukyus, Okinawa 903-0129, Japan Department of Mathematics, University of the Ryukyus, Okinawa 903-0129, Japan
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online 9 March 2012 Communicated by P.R. Holland
Some questions on the generalized Hartman effect presented by Kudaka and Matsumoto [S. Kudaka, S. Matsumoto, Phys. Lett. A 375 (2011) 3259] and a comment on them given by Milanovic´ and Radovanovic´ are discussed. © 2012 Elsevier B.V. All rights reserved.
Keywords: Tunneling time Generalized Hartman effect Stationary phase method Superluminality
In 2002, Olkhovsky, Recami and Salesi studied the phenomenon of tunneling through two successive barriers separated by an intermediate free region [1]. They reported that the total tunneling time is independent not only of the width of both barriers, but also of the length of the free region. This surprising phenomenon has been called the “generalized Hartman effect.” In a previous paper [2], we focused our attention on the stationary phase method (SPM) which played an essential role in the reasoning in [1], and derived some conclusions: 1. A necessary mathematical condition for applying the SPM is not satisfied in its application in [1]. 2. Its correct application tells us that the transmitted wave is a sum of infinite wave packets each of which has a single peak: (a) If the length of the free region (denoted as L − a) is sufficiently large such that
( L − a)/ v + τ 1/δ,
(36) in [2]
then the effect of interference among these components is so small or negligible that the transmitted wave reveals itself as a wave with multiple peaks. (Here, v and δ denote the group velocity and the energy uncertainty of the incident packet respectively, and τ the tunneling time of a particle tunneling through one barrier.) Further, the time at which each peak departs from the second barrier does in fact depend linearly on the length L − a.
*
DOI of original article: 10.1016/j.physleta.2011.07.036. DOI of comment: 10.1016/j.physleta.2012.03.020. Corresponding author. E-mail address: [email protected] (S. Matsumoto).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2012.03.019
(b) In contrast, if L − a is rather small and the condition (36) is not satisfied, then the sum of the infinite wave packets can even be a wave with a single peak. (See Eq. (37) and the explanations following it.) Milanovic´ and Radovanovic´ [3] emphasize in their comment on our paper that if the energy uncertainty δ is sufficiently small then the transmitted wave has a single peak and the total tunneling time (i.e. the time at which the peak departs from the second barrier) does not depend on L − a. Their discussions are for the case in which the condition (36) is not satisfied, and we agree with their result mentioned above. We certainly lacked in our Letter the statement indicated above in italics. Their comment is very important for us, because it indicates that they have agreed with us on the point that the generalized Hartman effect appears only when the length of the free region is rather small and it disappears when the length becomes very large.1 An implication of our agreement is that we should be cautious when we define the “velocity” of the particle as the combined width of the double-barrier structure divided by the total tunneling time. If we try, for example, to argue about large “velocity” by extending L − a, we have to make the energy width δ decrease, and 1 The particle has a time uncertainty δt ≈ 1/δ and a position uncertainty δ x ≈ v δt. The condition (36) means therefore that the free region is much wider than the coordinate width of the wave packet. That is to say, this condition means that the position of the particle can be specified much accurately between the two barriers. In this case, it is very natural to imagine that there exist multiple reflections between those barriers. As De Leo and Rotelli stated in [4], it is quite strange to admit those multiple reflections without admitting multiple outgoing wave packets.
1404
S. Kudaka, S. Matsumoto / Physics Letters A 376 (2012) 1403–1404
consequently, have to make the time uncertainty δt ≈ 1/δ increase. The time uncertainty δt is the precision of measurement to determine the time at which the particle passes a given point [5]. Therefore, for a particle with a very large δt, we face circumstances under which we should cautiously consider what physical reality we can attach to this “velocity.” Similar circumstances occur with a particle tunneling through a single barrier. Hartman told us on page 3433 in [6] that the transmission for a very thick barrier is over the top of the barrier for the most part, essentially not tunneling, and that the tunneling time (= δt 3 in his terminology) in this case depends linearly on the width (= a) of the barrier. The upper limit of the width a over which his effect disappears is found to be 1/δ up to a multiplicative constant, where δ is the energy uncertainty of the incident particle.2 The value a/δt 3 can be arbitrarily large if we make δ small, and can even be larger than the speed of light. But then, the time uncertainty δt ≈ 1/δ increases and allows a possible overlap of entry and exit. The arrival time of the particle cannot be specified more accurately than an interval (−δt , δt ) if we try not to disturb the state of the particle. And the departure time of that particle can-
2 If we denote the height of the barrier and the mean energy of the incident √ particle as V 0 and E 0 , the threshhold is given by ( V 0 − E 0 )/m(1/δ) up to a numerical multiplier of order 1, where m is the mass of the particle. (In relativistic √ case, it is given by ( V 0 + m − E 0 )/m(1/δ).)
not be specified to any interval narrower than (δt 3 − δt , δt 3 + δt ). When δt is very large, these intervals overlap each other extensively.3 We follow here a standard interpretation of quantum mechanics [7] that if there is no measurement process to confirm the existence of a physical characteristic then we cannot attach any reality to it. Hence, for a particle with a very large δt, we cannot state that this particle arrives at one end of the barrier once and departs from the other end after a lapse of time δt 3 . We should cautiously consider what physical realities we can attach to the values δt 3 and a/δt 3 . References [1] [2] [3] [4] [5] [6] [7]
3
V.S. Olkhovsky, E. Recami, G. Salesi, Europhys. Lett. 57 (2002) 879. S. Kudaka, S. Matsumoto, Phys. Lett. A 375 (2011) 3259. ´ J. Ranovanovic, ´ Phys. Lett. A 376 (16) (2012) 1401. V. Milanovic, S. De Leo, P.R. Rotelli, Phys. Lett. A 342 (2005) 294. A. Messiah, Mécanique Quantique, Dunod, Paris, 1959. T.E. Hartman, J. Appl. Phys. 33 (1962) 3427. N. Bohr, Atomic Theory and the Description of Nature, Cambridge Univ. Press, Cambridge, 1934.
The extent of this overlap and the value a/δt 3 have a relation that
δt /δt 3
m/( V 0 − E 0 )(a/δt 3 ).
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Comment
Reply to “Comment on: ‘Questions concerning the generalized Hartman effect’ [Phys. Lett. A 375 (2011) 3259]” Shoju Kudaka a , Shuichi Matsumoto b,∗ a b
Department of Physics, University of the Ryukyus, Okinawa 903-0129, Japan Department of Mathematics, University of the Ryukyus, Okinawa 903-0129, Japan
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online 9 March 2012 Communicated by P.R. Holland
Some questions on the generalized Hartman effect presented by Kudaka and Matsumoto [S. Kudaka, S. Matsumoto, Phys. Lett. A 375 (2011) 3259] and a comment on them given by Milanovic´ and Radovanovic´ are discussed. © 2012 Elsevier B.V. All rights reserved.
Keywords: Tunneling time Generalized Hartman effect Stationary phase method Superluminality
In 2002, Olkhovsky, Recami and Salesi studied the phenomenon of tunneling through two successive barriers separated by an intermediate free region [1]. They reported that the total tunneling time is independent not only of the width of both barriers, but also of the length of the free region. This surprising phenomenon has been called the “generalized Hartman effect.” In a previous paper [2], we focused our attention on the stationary phase method (SPM) which played an essential role in the reasoning in [1], and derived some conclusions: 1. A necessary mathematical condition for applying the SPM is not satisfied in its application in [1]. 2. Its correct application tells us that the transmitted wave is a sum of infinite wave packets each of which has a single peak: (a) If the length of the free region (denoted as L − a) is sufficiently large such that
( L − a)/ v + τ 1/δ,
(36) in [2]
then the effect of interference among these components is so small or negligible that the transmitted wave reveals itself as a wave with multiple peaks. (Here, v and δ denote the group velocity and the energy uncertainty of the incident packet respectively, and τ the tunneling time of a particle tunneling through one barrier.) Further, the time at which each peak departs from the second barrier does in fact depend linearly on the length L − a.
*
DOI of original article: 10.1016/j.physleta.2011.07.036. DOI of comment: 10.1016/j.physleta.2012.03.020. Corresponding author. E-mail address: [email protected] (S. Matsumoto).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2012.03.019
(b) In contrast, if L − a is rather small and the condition (36) is not satisfied, then the sum of the infinite wave packets can even be a wave with a single peak. (See Eq. (37) and the explanations following it.) Milanovic´ and Radovanovic´ [3] emphasize in their comment on our paper that if the energy uncertainty δ is sufficiently small then the transmitted wave has a single peak and the total tunneling time (i.e. the time at which the peak departs from the second barrier) does not depend on L − a. Their discussions are for the case in which the condition (36) is not satisfied, and we agree with their result mentioned above. We certainly lacked in our Letter the statement indicated above in italics. Their comment is very important for us, because it indicates that they have agreed with us on the point that the generalized Hartman effect appears only when the length of the free region is rather small and it disappears when the length becomes very large.1 An implication of our agreement is that we should be cautious when we define the “velocity” of the particle as the combined width of the double-barrier structure divided by the total tunneling time. If we try, for example, to argue about large “velocity” by extending L − a, we have to make the energy width δ decrease, and 1 The particle has a time uncertainty δt ≈ 1/δ and a position uncertainty δ x ≈ v δt. The condition (36) means therefore that the free region is much wider than the coordinate width of the wave packet. That is to say, this condition means that the position of the particle can be specified much accurately between the two barriers. In this case, it is very natural to imagine that there exist multiple reflections between those barriers. As De Leo and Rotelli stated in [4], it is quite strange to admit those multiple reflections without admitting multiple outgoing wave packets.
1404
S. Kudaka, S. Matsumoto / Physics Letters A 376 (2012) 1403–1404
consequently, have to make the time uncertainty δt ≈ 1/δ increase. The time uncertainty δt is the precision of measurement to determine the time at which the particle passes a given point [5]. Therefore, for a particle with a very large δt, we face circumstances under which we should cautiously consider what physical reality we can attach to this “velocity.” Similar circumstances occur with a particle tunneling through a single barrier. Hartman told us on page 3433 in [6] that the transmission for a very thick barrier is over the top of the barrier for the most part, essentially not tunneling, and that the tunneling time (= δt 3 in his terminology) in this case depends linearly on the width (= a) of the barrier. The upper limit of the width a over which his effect disappears is found to be 1/δ up to a multiplicative constant, where δ is the energy uncertainty of the incident particle.2 The value a/δt 3 can be arbitrarily large if we make δ small, and can even be larger than the speed of light. But then, the time uncertainty δt ≈ 1/δ increases and allows a possible overlap of entry and exit. The arrival time of the particle cannot be specified more accurately than an interval (−δt , δt ) if we try not to disturb the state of the particle. And the departure time of that particle can-
2 If we denote the height of the barrier and the mean energy of the incident √ particle as V 0 and E 0 , the threshhold is given by ( V 0 − E 0 )/m(1/δ) up to a numerical multiplier of order 1, where m is the mass of the particle. (In relativistic √ case, it is given by ( V 0 + m − E 0 )/m(1/δ).)
not be specified to any interval narrower than (δt 3 − δt , δt 3 + δt ). When δt is very large, these intervals overlap each other extensively.3 We follow here a standard interpretation of quantum mechanics [7] that if there is no measurement process to confirm the existence of a physical characteristic then we cannot attach any reality to it. Hence, for a particle with a very large δt, we cannot state that this particle arrives at one end of the barrier once and departs from the other end after a lapse of time δt 3 . We should cautiously consider what physical realities we can attach to the values δt 3 and a/δt 3 . References [1] [2] [3] [4] [5] [6] [7]
3
V.S. Olkhovsky, E. Recami, G. Salesi, Europhys. Lett. 57 (2002) 879. S. Kudaka, S. Matsumoto, Phys. Lett. A 375 (2011) 3259. ´ J. Ranovanovic, ´ Phys. Lett. A 376 (16) (2012) 1401. V. Milanovic, S. De Leo, P.R. Rotelli, Phys. Lett. A 342 (2005) 294. A. Messiah, Mécanique Quantique, Dunod, Paris, 1959. T.E. Hartman, J. Appl. Phys. 33 (1962) 3427. N. Bohr, Atomic Theory and the Description of Nature, Cambridge Univ. Press, Cambridge, 1934.
The extent of this overlap and the value a/δt 3 have a relation that
δt /δt 3
m/( V 0 − E 0 )(a/δt 3 ).