~ a : Since the energy of the ground state is known, this argument can be simplified. Turning point is twice off radius be four one s state The probability that electron is it classical forward A region is probability p are greater than to wait Toby equal toe. Textbook solution for Introduction To Quantum Mechanics 3rd Edition Griffiths Chapter 2.3 Problem 2.14P. (a) Determine the expectation value of . [3] P. W. Atkins, J. de Paula, and R. S. Friedman, Quanta, Matter and Change: A Molecular Approach to Physical Chemistry, New York: Oxford University Press, 2009 p. 66. << #k3 b[5Uve. hb \(0Ik8>k!9h 2K-y!wc' (Z[0ma7m#GPB0F62:b In the same way as we generated the propagation factor for a classically . A corresponding wave function centered at the point x = a will be . endobj /Contents 10 0 R (4.172), \psi _{n}(x)=1/\sqrt{\sqrt{\pi }2^{n}n!x_{0} } e^{-x^{2} /2x^{2}_{0}}H_{n}(x/x_{0}), where x_{0} is given by x_{0}=\sqrt{\hbar /(m\omega )}. << /S /GoTo /D [5 0 R /Fit] >> Such behavior is strictly forbidden in classical mechanics, according to which a particle of energy is restricted to regions of space where (Fitzpatrick 2012). h 1=4 e m!x2=2h (1) The probability that the particle is found between two points aand bis P ab= Z b a 2 0(x)dx (2) so the probability that the particle is in the classical region is P . Find step-by-step Physics solutions and your answer to the following textbook question: In the ground state of the harmonic oscillator, what is the probability (correct to three significant digits) of finding the particle outside the classically allowed region? Recovering from a blunder I made while emailing a professor. Jun Thus, the particle can penetrate into the forbidden region. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. He killed by foot on simplifying. E < V . Calculate the classically allowed region for a particle being in a one-dimensional quantum simple harmonic energy eigenstate |n). quantum-mechanics 7 0 obj Quantum tunneling through a barrier V E = T . This property of the wave function enables the quantum tunneling. The answer is unfortunately no. The oscillating wave function inside the potential well dr(x) 0.3711, The wave functions match at x = L Penetration distance Classically forbidden region tance is called the penetration distance: Year . The part I still get tripped up on is the whole measuring business. We have so far treated with the propagation factor across a classically allowed region, finding that whether the particle is moving to the left or the right, this factor is given by where a is the length of the region and k is the constant wave vector across the region. << (B) What is the expectation value of x for this particle? For the particle to be found with greatest probability at the center of the well, we expect . /MediaBox [0 0 612 792] 6 0 obj Title . The integral you wrote is the probability of being betwwen $a$ and $b$, Sorry, I misunderstood the question. The connection of the two functions means that a particle starting out in the well on the left side has a finite probability of tunneling through the barrier and being found on the right side even though the energy of the particle is less than the barrier height. But for the quantum oscillator, there is always a nonzero probability of finding the point in a classically forbidden region; in other words, there is a nonzero tunneling probability. \int_{\sqrt{2n+1} }^{+\infty }e^{-y^{2}}H^{2}_{n}(x) dy, (4.298). We need to find the turning points where En. (iv) Provide an argument to show that for the region is classically forbidden. You may assume that has been chosen so that is normalized. If the proton successfully tunnels into the well, estimate the lifetime of the resulting state. The classically forbidden region!!! Its deviation from the equilibrium position is given by the formula. (a) Determine the probability of finding a particle in the classically forbidden region of a harmonic oscillator for the states n=0, 1, 2, 3, 4. The probability of finding the particle in an interval x about the position x is equal to (x) 2 x. Open content licensed under CC BY-NC-SA, Think about a classical oscillator, a swing, a weight on a spring, a pendulum in a clock. >> .1b[K*Tl&`E^,;zmH4(2FtS> xZDF4:mj mS%\klB4L8*H5%*@{N Probability of particle being in the classically forbidden region for the simple harmonic oscillator: a. Quantum mechanically, there exist states (any n > 0) for which there are locations x, where the probability of finding the particle is zero, and that these locations separate regions of high probability! So that turns out to be scared of the pie. >> .r#+_. \[ \delta = \frac{\hbar c}{\sqrt{8mc^2(U-E)}}\], \[\delta = \frac{197.3 \text{ MeVfm} }{\sqrt{8(938 \text{ MeV}}}(20 \text{ MeV -10 MeV})\]. endobj Energy and position are incompatible measurements. A particle has a certain probability of being observed inside (or outside) the classically forbidden region, and any measurements we make . classically forbidden region: Tunneling . There are numerous applications of quantum tunnelling. probability of finding particle in classically forbidden region. Classically, there is zero probability for the particle to penetrate beyond the turning points and . I am not sure you could even describe it as being a particle when it's inside the barrier, the wavefunction is evanescent (decaying). The classically forbidden region is shown by the shading of the regions beyond Q0 in the graph you constructed for Exercise \(\PageIndex{26}\). Legal. Turning point is twice off radius be four one s state The probability that electron is it classical forward A region is probability p are greater than to wait Toby equal toe. Are there any experiments that have actually tried to do this? << Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. What is the probability of finding the particle in classically forbidden region in ground state of simple harmonic oscillator. =gmrw_kB!]U/QVwyMI: Connect and share knowledge within a single location that is structured and easy to search. We should be able to calculate the probability that the quantum mechanical harmonic oscillator is in the classically forbidden region for the lowest energy state, the state with v = 0. in thermal equilibrium at (kelvin) Temperature T the average kinetic energy of a particle is . 2. Or since we know it's kinetic energy accurately because of HUP I can't say anything about its position? Connect and share knowledge within a single location that is structured and easy to search. calculate the probability of nding the electron in this region. How to notate a grace note at the start of a bar with lilypond? Can you explain this answer? /Length 2484 (ZapperZ's post that he linked to describes experiments with superconductors that show that interactions can take place within the barrier region, but they still don't actually measure the particle's position to be within the barrier region.). There is nothing special about the point a 2 = 0 corresponding to the "no-boundary proposal". This superb text by David Bohm, formerly Princeton University and Emeritus Professor of Theoretical Physics at Birkbeck College, University of London, provides a formulation of the quantum theory in terms of qualitative and imaginative concepts that have evolved outside and beyond classical theory. Is it possible to create a concave light? Mesoscopic and microscopic dipole clusters: Structure and phase transitions A.I. endobj Peter, if a particle can be in a classically forbidden region (by your own admission) why can't we measure/detect it there? c What is the probability of finding the particle in the classically forbidden from PHYSICS 202 at Zewail University of Science and Technology L2 : Classical Approach - Probability , Maths, Class 10; Video | 09:06 min. You can't just arbitrarily "pick" it to be there, at least not in any "ordinary" cases of tunneling, because you don't control the particle's motion. Find the Source, Textbook, Solution Manual that you are looking for in 1 click. /Filter /FlateDecode The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Non-zero probability to . Although it presents the main ideas of quantum theory essentially in nonmathematical terms, it . 24 0 obj The classically forbidden region is given by the radial turning points beyond which the particle does not have enough kinetic energy to be there (the kinetic energy would have to be negative). 2003-2023 Chegg Inc. All rights reserved. Thus, the energy levels are equally spaced starting with the zero-point energy hv0 (Fig. You've requested a page on a website (ftp.thewashingtoncountylibrary.com) that is on the Cloudflare network. represents a single particle then 2 called the probability density is the from PHY 1051 at Manipal Institute of Technology If the particle penetrates through the entire forbidden region, it can "appear" in the allowed region x > L. Such behavior is strictly forbidden in classical mechanics, according to which a particle of energy is restricted to regions of space where (Fitzpatrick 2012). By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. The classically forbidden region coresponds to the region in which $$ T (x,t)=E (t)-V (x) <0$$ in this case, you know the potential energy $V (x)=\displaystyle\frac {1} {2}m\omega^2x^2$ and the energy of the system is a superposition of $E_ {1}$ and $E_ {3}$. /Type /Annot +!_u'4Wu4a5AkV~NNl 15-A3fLF[UeGH5Fc. Lehigh Course Catalog (1996-1997) Date Created . Using indicator constraint with two variables. The integral in (4.298) can be evaluated only numerically. %PDF-1.5 Particle Properties of Matter Chapter 14: 7. It only takes a minute to sign up. (v) Show that the probability that the particle is found in the classically forbidden region is and that the expectation value of the kinetic energy is . Particles in classically forbidden regions E particle How far does the particle extend into the forbidden region? Misterio Quartz With White Cabinets, Harmonic . The wave function in the classically forbidden region of a finite potential well is The wave function oscillates until it reaches the classical turning point at x = L, then it decays exponentially within the classically forbidden region. quantum mechanics; jee; jee mains; Share It On Facebook Twitter Email . The probability of finding a ground-state quantum particle in the classically forbidden region is about 16%. HOME; EVENTS; ABOUT; CONTACT; FOR ADULTS; FOR KIDS; tonya francisco biography 2. . Take advantage of the WolframNotebookEmebedder for the recommended user experience. /Annots [ 6 0 R 7 0 R 8 0 R ] Take the inner products. This shows that the probability decreases as n increases, so it would be very small for very large values of n. It is therefore unlikely to find the particle in the classically forbidden region when the particle is in a very highly excited state. Why does Mister Mxyzptlk need to have a weakness in the comics? dq represents the probability of finding a particle with coordinates q in the interval dq (assuming that q is a continuous variable, like coordinate x or momentum p). Thus, there is about a one-in-a-thousand chance that the proton will tunnel through the barrier. Last Post; Jan 31, 2020; Replies 2 Views 880. /Border[0 0 1]/H/I/C[0 1 1] a is a constant. June 23, 2022 What sort of strategies would a medieval military use against a fantasy giant? Is it just hard experimentally or is it physically impossible? For a quantum oscillator, we can work out the probability that the particle is found outside the classical region. 25 0 obj For the n = 1 state calculate the probability that the particle will be found in the classically forbidden region. When the width L of the barrier is infinite and its height is finite, a part of the wave packet representing . endstream = h 3 m k B T . The same applies to quantum tunneling. If the correspondence principle is correct the quantum and classical probability of finding a particle in a particular position should approach each other for very high energies. If the particle penetrates through the entire forbidden region, it can appear in the allowed region x > L. This is referred to as quantum tunneling and illustrates one of the most fundamental distinctions between the classical and quantum worlds. Accueil; Services; Ralisations; Annie Moussin; Mdias; 514-569-8476 . We have step-by-step solutions for your textbooks written by Bartleby experts! "`Z@,,Y.$U^,' N>w>j4'D$(K$`L_rhHn_\^H'#k}_GWw>?=Q1apuOW0lXiDNL!CwuY,TZNg#>1{lpUXHtFJQ9""x:]-V??e 9NoMG6^|?o.d7wab=)y8u}m\y\+V,y C ~ 4K5,,>h!b$,+e17Wi1g_mef~q/fsx=a`B4("B&oi; Gx#b>Lx'$2UDPftq8+<9`yrs W046;2P S --66 ,c0$?2 QkAe9IMdXK \W?[ 4\bI'EXl]~gr6 q 8d$ $,GJ,NX-b/WyXSm{/65'*kF{>;1i#CC=`Op l3//BC#!!Z 75t`RAH$H @ )dz/)y(CZC0Q8o($=guc|A&!Rxdb*!db)d3MV4At2J7Xf2e>Yb )2xP'gHH3iuv AkZ-:bSpyc9O1uNFj~cK\y,W-_fYU6YYyU@6M^ nu#)~B=jDB5j?P6.LW:8X!NhR)da3U^w,p%} u\ymI_7 dkHgP"v]XZ A)r:jR-4,B endobj Year . But for the quantum oscillator, there is always a nonzero probability of finding the point in a classically forbidden region; in other words, there is a nonzero tunneling probability. A particle has a probability of being in a specific place at a particular time, and this probabiliy is described by the square of its wavefunction, i.e | ( x, t) | 2. Qfe lG+,@#SSRt!(` 9[bk&TczF4^//;SF1-R;U^SN42gYowo>urUe\?_LiQ]nZh \[\delta = \frac{1}{2\alpha}\], \[\delta = \frac{\hbar x}{\sqrt{8mc^2 (U-E)}}\], The penetration depth defines the approximate distance that a wavefunction extends into a forbidden region of a potential. Acidity of alcohols and basicity of amines. Quantum mechanics, with its revolutionary implications, has posed innumerable problems to philosophers of science. It came from the many worlds , , you see it moves throw ananter dimension ( some kind of MWI ), I'm having trouble wrapping my head around the idea of a particle being in a classically prohibited region. Posted on . /ProcSet [ /PDF /Text ] Seeing that ^2 in not nonzero inside classically prohibited regions, could we theoretically detect a particle in a classically prohibited region? (vtq%xlv-m:'yQp|W{G~ch iHOf>Gd*Pv|*lJHne;(-:8!4mP!.G6stlMt6l\mSk!^5@~m&D]DkH[*. June 5, 2022 . This dis- FIGURE 41.15 The wave function in the classically forbidden region. Q) Calculate for the ground state of the hydrogen atom the probability of finding the electron in the classically forbidden region. Share Cite Why is the probability of finding a particle in a quantum well greatest at its center? Have particles ever been found in the classically forbidden regions of potentials? The probability of that is calculable, and works out to 13e -4, or about 1 in 4. Euler: A baby on his lap, a cat on his back thats how he wrote his immortal works (origin? (1) A sp. Classically, there is zero probability for the particle to penetrate beyond the turning points and . Mount Prospect Lions Club Scholarship, Description . Reuse & Permissions Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. If so, how close was it? If not, isn't that inconsistent with the idea that (x)^2dx gives us the probability of finding a particle in the region of x-x+dx? Making statements based on opinion; back them up with references or personal experience. You don't need to take the integral : you are at a situation where $a=x$, $b=x+dx$. They have a certain characteristic spring constant and a mass. endobj Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. But for the quantum oscillator, there is always a nonzero probability of finding the point in a classically forbidden region; in other words, there is a nonzero tunneling probability. In a crude approximation of a collision between a proton and a heavy nucleus, imagine an 10 MeV proton incident on a symmetric potential well of barrier height 20 MeV, barrier width 5 fm, well depth -50 MeV, and well width 15 fm. The answer would be a yes. These regions are referred to as allowed regions because the kinetic energy of the particle (KE = E U) is a real, positive value. Can a particle be physically observed inside a quantum barrier? So the forbidden region is when the energy of the particle is less than the . where the Hermite polynomials H_{n}(y) are listed in (4.120). /Rect [154.367 463.803 246.176 476.489] I do not see how, based on the inelastic tunneling experiments, one can still have doubts that the particle did, in fact, physically traveled through the barrier, rather than simply appearing at the other side. MathJax reference. To me, this would seem to imply negative kinetic energy (and hence imaginary momentum), if we accept that total energy = kinetic energy + potential energy. How to match a specific column position till the end of line? L2 : Classical Approach - Probability , Maths, Class 10; Video | 09:06 min. Is it just hard experimentally or is it physically impossible? It is the classically allowed region (blue). Experts are tested by Chegg as specialists in their subject area. Not very far! A particle in an infinitely deep square well has a wave function given by ( ) = L x L x 2 2 sin. Textbook solution for Introduction To Quantum Mechanics 3rd Edition Griffiths Chapter 2.3 Problem 2.14P. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. in the exponential fall-off regions) ? . rev2023.3.3.43278. Performance & security by Cloudflare. Each graph is scaled so that the classical turning points are always at and . We have step-by-step solutions for your textbooks written by Bartleby experts! This made sense to me but then if this is true, tunneling doesn't really seem as mysterious/mystifying as it was presented to be. You can see the sequence of plots of probability densities, the classical limits, and the tunneling probability for each . Step by step explanation on how to find a particle in a 1D box. "After the incident", I started to be more careful not to trip over things. Slow down electron in zero gravity vacuum. A typical measure of the extent of an exponential function is the distance over which it drops to 1/e of its original value. Has a particle ever been observed while tunneling? In the present work, we shall also study a 1D model but for the case of the long-range soft-core Coulomb potential. Belousov and Yu.E. In this approximation of nuclear fusion, an incoming proton can tunnel into a pre-existing nuclear well. Using indicator constraint with two variables. endobj Here's a paper which seems to reflect what some of what the OP's TA was saying (and I think Vanadium 50 too). The probability of finding a ground-state quantum particle in the classically forbidden region is about 16%. For the harmonic oscillator in it's ground state show the probability of fi, The probability of finding a particle inside the classical limits for an os, Canonical Invariants, Harmonic Oscillator. Surly Straggler vs. other types of steel frames. /Rect [396.74 564.698 465.775 577.385] >> "Quantum Harmonic Oscillator Tunneling into Classically Forbidden Regions", http://demonstrations.wolfram.com/QuantumHarmonicOscillatorTunnelingIntoClassicallyForbiddenRe/, Time Evolution of Squeezed Quantum States of the Harmonic Oscillator, Quantum Octahedral Fractal via Random Spin-State Jumps, Wigner Distribution Function for Harmonic Oscillator, Quantum Harmonic Oscillator Tunneling into Classically Forbidden Regions. All that remains is to determine how long this proton will remain in the well until tunneling back out. << Perhaps all 3 answers I got originally are the same? There is also a U-shaped curve representing the classical probability density of finding the swing at a given position given only its energy, independent of phase. (iv) Provide an argument to show that for the region is classically forbidden. \[ \Psi(x) = Ae^{-\alpha X}\] Particle always bounces back if E < V . So, if we assign a probability P that the particle is at the slit with position d/2 and a probability 1 P that it is at the position of the slit at d/2 based on the observed outcome of the measurement, then the mean position of the electron is now (x) = Pd/ 2 (1 P)d/ 2 = (P 1 )d. and the standard deviation of this outcome is Also, note that there is appreciable probability that the particle can be found outside the range , where classically it is strictly forbidden! . Zoning Sacramento County, Can you explain this answer? /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R E is the energy state of the wavefunction. This is impossible as particles are quantum objects they do not have the well defined trajectories we are used to from Classical Mechanics. Go through the barrier . Can you explain this answer?, a detailed solution for What is the probability of finding the particle in classically forbidden region in ground state of simple harmonic oscillatorCorrect answer is '0.18'. in English & in Hindi are available as part of our courses for Physics. sage steele husband jonathan bailey ng nhp/ ng k . Beltway 8 Accident This Morning, For a quantum oscillator, assuming units in which Planck's constant , the possible values of energy are no longer a continuum but are quantized with the possible values: . Using this definition, the tunneling probability (T), the probability that a particle can tunnel through a classically impermeable barrier, is given by /Font << /F85 13 0 R /F86 14 0 R /F55 15 0 R /F88 16 0 R /F92 17 0 R /F93 18 0 R /F56 20 0 R /F100 22 0 R >> Third, the probability density distributions | n (x) | 2 | n (x) | 2 for a quantum oscillator in the ground low-energy state, 0 (x) 0 (x), is largest at the middle of the well (x = 0) (x = 0). 5 0 obj Mathematically this leads to an exponential decay of the probability of finding the particle in the classically forbidden region, i.e. $x$-representation of half (truncated) harmonic oscillator? for 0 x L and zero otherwise. Use MathJax to format equations. The probability of the particle to be found at position x at time t is calculated to be $\left|\psi\right|^2=\psi \psi^*$ which is $\sqrt {A^2 (\cos^2+\sin^2)}$. You simply cannot follow a particle's trajectory because quite frankly such a thing does not exist in Quantum Mechanics. To find the probability amplitude for the particle to be found in the up state, we take the inner product for the up state and the down state. Is a PhD visitor considered as a visiting scholar? In that work, the details of calculation of probability distributions of tunneling times were presented for the case of half-cycle pulse and when ionization occurs completely by tunneling (from classically forbidden region). \int_{\sqrt{3} }^{\infty }y^{2}e^{-y^{2}}dy=0.0495. xZrH+070}dHLw 06*T Y+i-a3"4 c where S (x) is the amplitude of waves at x that originated from the source S. This then is the probability amplitude of observing a particle at x given that it originated from the source S , i. by the Born interpretation Eq. - the incident has nothing to do with me; can I use this this way? Although the potential outside of the well is due to electric repulsion, which has the 1/r dependence shown below. Ok. Kind of strange question, but I think I know what you mean :) Thank you very much. The same applies to quantum tunneling. endobj I think I am doing something wrong but I know what! 21 0 obj Can you explain this answer? /D [5 0 R /XYZ 200.61 197.627 null] See Answer please show step by step solution with explanation find the particle in the . It might depend on what you mean by "observe". [2] B. Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena, New York: Springer, 2000 p. 168. Powered by WOLFRAM TECHNOLOGIES << Correct answer is '0.18'. This is what we expect, since the classical approximation is recovered in the limit of high values . 9 0 obj For simplicity, choose units so that these constants are both 1. This is . Peter, if a particle can be in a classically forbidden region (by your own admission) why can't we measure/detect it there? I view the lectures from iTunesU which does not provide me with a URL. A similar analysis can be done for x 0. << where is a Hermite polynomial. Forbidden Region. Wavepacket may or may not . For a classical oscillator, the energy can be any positive number. \[ \tau = \bigg( \frac{15 x 10^{-15} \text{ m}}{1.0 x 10^8 \text{ m/s}}\bigg)\bigg( \frac{1}{0.97 x 10^{-3}} \]. Can you explain this answer? Once in the well, the proton will remain for a certain amount of time until it tunnels back out of the well. b. Contributed by: Arkadiusz Jadczyk(January 2015) \[T \approx 0.97x10^{-3}\] This is referred to as a forbidden region since the kinetic energy is negative, which is forbidden in classical physics. The best answers are voted up and rise to the top, Not the answer you're looking for? Find the probabilities of the state below and check that they sum to unity, as required. Well, let's say it's going to first move this way, then it's going to reach some point where the potential causes of bring enough force to pull the particle back towards the green part, the green dot and then its momentum is going to bring it past the green dot into the up towards the left until the force is until the restoring force drags the . ross university vet school housing. S>|lD+a +(45%3e;A\vfN[x0`BXjvLy. y_TT`/UL,v] (4), S (x) 2 dx is the probability density of observing a particle in the region x to x + dx. (a) Show by direct substitution that the function, An attempt to build a physical picture of the Quantum Nature of Matter Chapter 16: Part II: Mathematical Formulation of the Quantum Theory Chapter 17: 9. Annie Moussin designer intrieur. The values of r for which V(r)= e 2 . My TA said that the act of measurement would impart energy to the particle (changing the in the process), thus allowing it to get over that barrier and be in the classically prohibited region and conserving energy in the process. Also, note that there is appreciable probability that the particle can be found outside the range , where classically it is strictly forbidden! . /D [5 0 R /XYZ 126.672 675.95 null] A particle has a certain probability of being observed inside (or outside) the classically forbidden region, and any measurements we make will only either observe a particle there or they will not observe it there. Go through the barrier . \int_{\sqrt{7} }^{\infty }(8y^{3}-12y)^{2}e^{-y^{2}}dy=3.6363. What is the probability of finding the partic 1 Crore+ students have signed up on EduRev. Energy eigenstates are therefore called stationary states . Either way, you can observe a particle inside the barrier and later outside the barrier but you can not observe whether it tunneled through or jumped over. The classically forbidden region is given by the radial turning points beyond which the particle does not have enough kinetic energy to be there (the kinetic energy would have to be negative). 1. But there's still the whole thing about whether or not we can measure a particle inside the barrier. . What video game is Charlie playing in Poker Face S01E07? However, the probability of finding the particle in this region is not zero but rather is given by: (6.7.2) P ( x) = A 2 e 2 a X Thus, the particle can penetrate into the forbidden region. Consider the square barrier shown above. For a better experience, please enable JavaScript in your browser before proceeding. Okay, This is the the probability off finding the electron bill B minus four upon a cube eight to the power minus four to a Q plus a Q plus. Tunneling probabilities equal the areas under the curve beyond the classical turning points (vertical red lines).
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