Uncertainty principle and superradiance

Wen-Xiang Chen∗ Department of Astronomy, School of Physics and Materials Science, GuangZhou University

Associate Professor Hasegawa Yuji of the Vienna University of Technology and Professor Masaaki Ozawa of Nagoya University and other scholars published empirical results against Heisenberg’s uncertainty principle on January 15, 2012.They got a measurement result with a smaller error than the Heisenberg uncertainty principle, which proved the measurement advocated by the Heisenberg uncertainty principle.This article follows the method I used to study superradiation and connects the uncertainty principle with the superradiation effect. When the boundary conditions are assumed and part of the wave function is decoupled, the classical form of the probability function is obtained, and the boundary conditions can be clearly substituted into the expression of the uncertainty principle, and we get a lower measurement limit of the uncertainty principle. To say the least, due to the limitation of the uncertainty principle, there are also restrictions on the value of y. In the mentioned literature, the extreme value of the uncertainty principle can be smaller, and the value range of y can be larger. Keywords: Uncertainty principle, superradiance, Wronskian determinant

I. INTRODUCTION

In quantum mechanics, the uncertainty principle states that the position and momentum of a particle can not be determined simultaneously. The less the uncertainty of position, the greater the uncertainty of momentum, and vice versa. It’s the same thing. The meaning of uncertainty is different for different situations. It can be the observer’s lack of a certain amount of information, a certain amount of measurement error, or a similar set of preparatory work. Statistical diffusion value of the system. In 1927, the Werner Heisenberg published a paper on the physical implications of quantum theory, kinematics, and mechanics, giving an initial heuristic account of this principle, it is hoped that the physical properties of simple quantum experiments can be qualitatively analyzed and expressed successfully. This principle is also known as uncertainty principle. Later that year, Earl Kennard formulated the uncertainty relationship between position and momentum in strict mathematical terms. Two years later, the Howard Robertson promotes Kennard’s relationship. Similar uncertainty relations exist between energy and time, angular momentum and angle, and other physical quantities. Since uncertainty principle is the fundamental theory of quantum mechanics, many general experiments often involve questions about it. Some experiments will specifically test this principle or a similar principle. For example, test digital phase uncertainty principle in superconducting or quantum optical systems. The research on the uncertainty principle

∗Electronic address: [email protected] 2 can be used to develop the low noise technique for the gravity wave interferometer. The approach to superradiation – originally known as the Klein paradox – goes back to Klein’s original paper, which first examined the existence of step Dirac equation. He points out that electron beams travelling in a region large enough to be a potential barrier V may not have the exponential damping expected for non relativistic quantum tunneling processes. In an attempt to understand whether the result is a spurious artifact of the step potential used by Klein, Sauter found that the nature of the process is independent of the details of the obstacle, although the probability of propagation decreases as the slope decreases. In 1931, Thuot first described the phenomenon as “The Klein paradox”. In 1941, Hong De further studied the charged scalar field and the Klein-Gordon equation. The results show that when the potential is strong enough, the step potential produces a pair of charged particles. Hund tried to get the same result with fermions, but failed. Remarkably, this result can be seen as a precursor to the modern quantum field theory results of Julian Schwinger and Hokin. The latter shows that spontaneous pairing is possible in the presence of strong electromagnetic fields and the gravitational fields of bosons and fermions. In fact, we know today that the “Old”Klein paradox is solved by the formation of particle-antiparticle pairs on the potential barrier, thus explaining the non-decaying transmission part. In 1972, Press and Teukolsky[15] proposed that it is possible to add a mirror to the outside of a black hole to make a black hole bomb (according to the current explanation, this is a scattering process involving classical mechanics and quantum mechanics[1,3,9, 10, 13, 14]). When a bosonic wave is impinging upon a rotating black hole, the wave reflected by the event horizon will be amplified if the wave frequency ω lies in the following superradiant regime[11, 12, 15] a 0 < ω < mΩH , ΩH = 2 2 , (1) r+ + a where m is azimuthal number of the bosonic wave mode, ΩH is the angular velocity of black hole horizon.This amplification is superradiant scattering. Therefore, through the superradiation process, the rotational energy of the black hole can be extracted. If there is a mirror between the black hole’s horizon and infinite space, the amplified wave will scatter back and forth and grow exponentially, which will cause the black hole’s superradiation to become unstable. Associate Professor Hasegawa Yuji of the Vienna University of Technology and Professor Masaaki Ozawa of Nagoya University and other scholars published empirical results against Heisenberg’s uncertainty principle on January 15, 2012[5]. They used two instruments to measure and calculate the spin angle of the neutrons, which was less than the uncertainty principle’s error, and the uncertainty principle proved uncertainty principle’s claim. This restriction is wrong. However, uncertainty principle is still correct because it is the inherent quantum nature of the particle. This article uses my approach to superradiation to link uncertainty principle to the effects of superradiation. I found that the measurement limit of the uncertainty principle can be much smaller under the effect of superradiation.

II. THE SUPERRADIATION EFFECT OF BOSON SCATTERING

We get the Klein-Gordon equation[8]

;µ Φ;µ = 0 , (2) 3

µ where we defined Φ;µ ≡ (∂µ − ieAµ)Φ and e is the charge of the scalar field.We get A = {A0(x), 0},and eA0(x)can be equal to µ(where µ is the mass).   0 as x → −∞ A0 → . (3)  V as x → +∞

With Φ = e−iωtf(x), which is determined by the ordinary differential equation

d2f + (ω − eA )2 f = 0 . (4) dx2 0

We see that particles coming from −∞ and scattering off the potential with reflection and transmission amplitudes R and T respectively. With these boundary conditions, the solution to behaves asymptotically as

iωx −iωx fin(x) = Ie + Re , x → −∞, (5)

ikx fin(x) = T e , x → +∞ (6) where k = ±(ω − eV ). To define the sign of ω and k we must look at the wave’s group velocity. We require ∂ω/∂k > 0, so that they travel from the left to the right in the x–direction and we take ω > 0.

The reflection coefficient and transmission coefficient depend on the specific shape of the potential A0. However one can easily show that the Wronskian

df˜ df˜ W = f˜ 2 − f˜ 1 , (7) 1 dx 2 dx ˜ ˜ between two independent solutions, f1 and f2, of is conserved. From the equation on the other hand, if f is a solution ∗ 2 2 ω−eV 2 then its complex conjugate f is another linearly independent solution. We find|R| = |I| − ω |T | .Thus,for 0 < ω < eV ,it is possible to have superradiant amplification of the reflected current, i.e, |R| > |I|. There are other potentials that can be completely resolved, which can also show superradiation explicitly.

III. THAT RESULT CONTAINED SOME CONCLUSIONS THAT VIOLATED UNCERTAINTY PRINCIPLE

The general relativity theory’s field equation is,

1 8πG R − g R = − T (8) µν 2 µv c4 µν

When Tµv = 0 in vacuum state,

Rµv = 0 (9)

Space-time of spherical coordinates is

1 dτ 2 = A(t, r)dt2 − B(t, r)dr2 + r2dθ2 + r2 sinθdφ2 (10) c2 4

If we use Eq(9),

A00 A0B0 A0 A02 B¨ B˙ 2 A˙B˙ R = − + − + + − − = 0 (11) tt 2B 4B2 Br 4AB 2B 4B2 4AB

A00 A02 A0B0 B0 B¨ A˙B˙ B˙ 2 R = − − − − + + = 0, (12) rr 2A 4A2 4AB Br 2A 4A2 4AB

1 rB0 rA0 B˙ R = −1 + − + = 0,R = R sin2 θ = 0,R = − = 0,R = R = R = R = R = 0 (13) θθ B 2B2 2AB φφ θθ tr Br tθ tφ rθ rφ θφ

0 ∂ 1 ∂ That time, = ∂r , · = c ∂t ,

B˙ = 0 (14)

We see that,

R R 1 A0 B0  (AB)0 tt + rr = − + = − = 0 (15) A B Br A B rAB2

Hence, we know that,

1 A = (16) B

If,

1 rB0 rA0  r 0 R = −1 + − + = −1 + = 0 (17) θθ B 2B2 2AB B

If we solve the Eq,

r 1 C = r + C → = 1 + (18) B B r

When r tends to infinity(r is a positive number. When near the event horizon, the above sign shows a negative number due to the tortoise coordinate), and we set C=ye−y, Therefore,

1 y A = = 1 − Σ, Σ = e−y (19) B r

 y X dτ 2 = 1 − dt2 (20) r

In this time, if particles’ mass are mi, the new energy is e,

2 2 2 2 E = Mc = m1c + m2c + ... + mnc + e (21)

We see that the boundary conditions of the preset fermions (the kind) are similar to ds space-time.At this time, positive pressure is generated in the flat spacetime. For[16]

ω ∗ 2 |hψA | ψAi| ≥ 1 − εA/2 (22) 5

The optimizer is of the form ω ω ∗ (1 − |α|) |ψAi + αA |ψAi |ψAi = , α ∈ [−1, 1] (23) cα

ω ∗ 1 |hψA | ψAi| = (1 − |α| + αA) (24) cα and

ω ∗ 1 2
|hψA|A|ψAi| = (1 − |α|)A + α A (25) cα This then proves the assertion. A general explanation of the uncertainty principle: 2 2 2 2 1 1 σ σ ≥ h{A, D}i − hAihDi + h[A, D]i (26) A D 2 2i If you multiply the formula by a number less than 1, then the extreme value of the uncertainty principle will become smaller. The principle of joint uncertainty shows that it is impossible to make joint measurement of position and momentum, that is, to measure position and momentum simultaneously, only approximate joint measurement can be made, and 2 2 ω−eV 2 the error follows the inequality ∆x∆p ≥ 1/2(in natural unit system).We find|R| = |I| − ω |T | ,and we know 2 ω−eV 2 that|R| ≥ − ω |T | is a necessary condition for the inequality ∆x∆p ≥ 1/2 to be established[5,8].We can pre-set the boundary conditions eA0(x) = yω[4], and we see that when y is relatively large(according to the properties of the 2 ω−eV 2 boson, y can be very large),|R| ≥ − ω |T | may not hold.In the end,we can get ∆x∆p ≥ 1/2 may not hold.When the boundary conditions are assumed and part of the wave function is decoupled, the classical form of the probability function is obtained, and the boundary conditions can be clearly substituted into the uncertainty principle expression, we get a lower uncertainty principle limit. To say the least, because of the limitation of the uncertainty principle, there is a limitation on the value of y, whereas in the literature above, the extreme value of uncertainty principle can be smaller, and the range of the value of y in turn is larger. Acknowledgements: We would like to thank Jing-Yi Zhang for generous help. This work is partially supported by National Natural Science Foundation of China(No. 11873025).

IV. SUMMARY

This article follows the method I used to study superradiation and connects the uncertainty principle with the superradiation effect. I found that under the superradiation effect, the measurement limit of the uncertainty principle can be smaller.As I mentioned above, for the decoupling of wave functions due to the preset boundary conditions, the extreme value of the uncertainty principle can be smaller, and vice versa, the extreme value of the uncertainty principle can be smaller, then the scope of our consideration of boundary conditions can be wider.

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