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Natural Science Forum / Physics / Research / January 2005Extremum principles - is there action below hbar/2?
I am starting to collect material for the new version of the motion mountain physics text. For this, I have started a discussion on the German newsgroup de.sci.physik which might be of interest also over here. My sparring partner there is Norbert Dragon, well-known for his excellent research in and teaching of physics. We are presently discussing the following issue: Is it possible in nature to measure action values below hbar/2? I claim that this is impossible; he is not so sure. Arguments in favour: - is part of Einstein-Brillouin-Keller quantization - no experimental evidence to the contrary - experimental evidence in favour (e.g. photons) - ecplains tunneling, particle decay, particle reactions Arguments against: - never heard of it, not in the books; - spin 0 particles do exist; spin and action has the same dimension, thus action values below hbar/2 are possible I would be especially interested in getting more arguments *against* the thesis. For example, I would be interested in any experiment (in whatever setting) that claims to have measured action values below hbar/2. I would also translate some of the points made on both sides. Christoph Schiller http://www.motionmountain.net > I am starting to collect material for the new version of the > motion mountain physics text. For this, I have started a discussion [quoted text clipped - 6 lines] > > Is it possible in nature to measure action values below hbar/2? It is not clear what is meant by "action value", but there are certainly some reasonable definitions which give a value of zero for quantum systems. First example: for Hamiltonian operator H, define the "action" A = integral dt <psi| H - i hbar del/del t |psi> . This is reasonable because variation with respect to |psi> (and <psi|) gives the Schroedinger equation (and its conjugate). Clearly, however, A = 0 for all solutions. Second example: first, for classical systems, define the "action" by the line integral A' = integral_trajectory dx p , for closed trajectories, where p denotes the momentum at position x. To carry over to quantum systems one needs a rule for determining p. An reasonable choice is p = grad S, where S is the phase of the wavefunction (eg, justify via deBroglie-Bohm interpretation of QM, which assigns precisely this momentum, or via continuity equation). However, for energy eigenstates one has p=0, and hence A' = 0 for such states. So please give a definition, formal or operational, of "action" in this context. of_1001_nights@hotmail.com wrote: > chri_schiller@yahoo.com wrote: > > Is it possible in nature to measure action values below hbar/2? This question is not answered by the comment, but it could lead to an answer. > It is not clear what is meant by "action value", but there are > certainly some reasonable definitions which give a value of zero for [quoted text clipped - 3 lines] > A = integral dt <psi| H - i hbar del/del t |psi> . > This is reasonable because variation with respect to |psi> (and <psi|) > gives the Schroedinger equation (and its conjugate). Clearly, however, > A = 0 > for all solutions. Yes. > Second example: first, for classical systems, define the "action" by > the line integral > A' = integral_trajectory dx p , EBK quantization shows that this action is (n+something/2) hbar, where the something is usually 1; more can be found in the December Issue of American Journal of Physics. > So please give a definition, formal or operational, of "action" in this > context. Well, in quantum theory action cannot be zero, because its classical limit is not zero. The only formal definition I know of is Schwinger's. He defines action W as W_12 = \int_t1^t2 L(t) dt with the Lagranigian operator L(t) defined (as usual) as L(t) = \Sum_n p_n (dq_n/dt) - H where H is the Hamiltonian, and p_n and q_n are the momenta and coordinates. With this definition of action, Schwinger shows that the equations of motion can be deduced from the quantum action principle: delta < initial | final > = (i/hbar) < initial | delta W_12 | final This is told in his posthumous book "Quantum Mechanics - Symbolism of Atomic Measurements". Now a task for you: is the action thus defined zero or non-zero in an energy eigenstate of the atom? Christoph Schiller of_1001_nights@hotmail.com wrote: > chri_schiller@yahoo.com wrote: > > Is it possible in nature to measure action values below hbar/2? This question is not answered by the comment, but it could lead to an answer. > It is not clear what is meant by "action value", but there are > certainly some reasonable definitions which give a value of zero for [quoted text clipped - 3 lines] > A = integral dt <psi| H - i hbar del/del t |psi> . > This is reasonable because variation with respect to |psi> (and <psi|) > gives the Schroedinger equation (and its conjugate). Clearly, however, > A = 0 > for all solutions. Yes. > Second example: first, for classical systems, define the "action" by > the line integral > A' = integral_trajectory dx p , EBK quantization shows that this action is (n+something/2) hbar, where the something is usually 1; more can be found in the December Issue of American Journal of Physics. > So please give a definition, formal or operational, of "action" in this > context. Well, in quantum theory action cannot be zero, because its classical limit is not zero. The only formal definition I know of is Schwinger's. He defines action W as W_12 = \int_t1^t2 L(t) dt with the Lagranigian operator L(t) defined (as usual) as L(t) = \Sum_n p_n (dq_n/dt) - H where H is the Hamiltonian, and p_n and q_n are the momenta and coordinates. With this definition of action, Schwinger shows that the equations of motion can be deduced from the quantum action principle: delta < initial | final > = (i/hbar) < initial | delta W_12 | final This is told in his posthumous book "Quantum Mechanics - Symbolism of Atomic Measurements". Now a task for you: is the action thus defined zero or non-zero in an energy eigenstate of the atom? Christoph Schiller > Is it possible in nature to measure action values below hbar/2? To make a long story, this is not possible. A new literature search showed that Bohr told about the "indivisibility" of the quantum of action hbar for many years, in all his seminars throughout the world. It then was a shock, with the discovery of spin, that the indivisible value was hbar/2 and not hbar. But the result remains. There is no action below hbar/2. Already Bohr had told that all effects of quantum theory follow from this indivisibility. His Como lecture is an example. After the spin/2 shock, these arguments somwehow were not much used any more, out of fear that even smaller values would be possible. However, hbar/2 really is the minimum value. > Arguments against: > - spin 0 particles do exist; spin and action has the same dimension, > thus action values below hbar/2 are possible. The action W is defined as W= \int L d phi, where L is the total angular momentum and phi the phase. The spin S enters in the total angular momentum together with orbital angular momentum. There is no way to have W smaller than hbar/2 for a particle of spin 0 that is composed, because the finite extension of such a particle always leads to a finite orbital angular momentum, and thus W is never 0. Spin 0 could lead to a zero W only if the particle is point-like, ie, elementary. However, no elementary particles of spin 0 are known. One is predicted to exist: the Higgs particle. The statement that hbar/2 is the smallest action in nature thus implies that the Higgs is composed, not elementary. (We will see in 2008, when the LHC in Geneva is ready.) (Except if there might be another way out of the contradiction...) > I would be especially interested in getting more arguments *against* > the thesis. No other attempts arrived yet. Regards Christoph Schiller |
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