endobj Find a probability of measuring energy E n. From (2.13) c n . Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. VwU|V5PbK\Y-O%!H{,5WQ_QC.UX,c72Ca#_R"n Performance & security by Cloudflare. Click to reveal http://demonstrations.wolfram.com/QuantumHarmonicOscillatorTunnelingIntoClassicallyForbiddenRe/ Classically, there is zero probability for the particle to penetrate beyond the turning points and . probability of finding particle in classically forbidden region. Mutually exclusive execution using std::atomic? Possible alternatives to quantum theory that explain the double slit experiment? = h 3 m k B T It may not display this or other websites correctly. endobj What video game is Charlie playing in Poker Face S01E07? Download more important topics, notes, lectures and mock test series for Physics Exam by signing up for free. 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 . 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. The part I still get tripped up on is the whole measuring business. endobj WEBVTT 00:00:00.060 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.800 Commons license. The wave function becomes a rather regular localized wave packet and its possible values of p and T are all non-negative. Share Cite By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Seeing that ^2 in not nonzero inside classically prohibited regions, could we theoretically detect a particle in a classically prohibited region? 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 . tests, examples and also practice Physics tests. The classical turning points are defined by [latex]E_{n} =V(x_{n} )[/latex] or by [latex]hbar omega (n+frac{1}{2} )=frac{1}{2}momega ^{2} The vibrational frequency of H2 is 131.9 THz. Experts are tested by Chegg as specialists in their subject area. So the forbidden region is when the energy of the particle is less than the . (That might tbecome a serious problem if the trend continues to provide content with no URLs), 2023 Physics Forums, All Rights Reserved, https://www.physicsforums.com/showpost.php?p=3063909&postcount=13, http://dx.doi.org/10.1103/PhysRevA.48.4084, http://en.wikipedia.org/wiki/Evanescent_wave, http://dx.doi.org/10.1103/PhysRevD.50.5409. 10 0 obj The turning points are thus given by . For the hydrogen atom in the first excited state, find the probability of finding the electron in a classically forbidden region. /Type /Annot Therefore, the probability that the particle lies outside the classically allowed region in the ground state is 1 a a j 0(x;t)j2 dx= 1 erf 1 0:157 . 23 0 obj :Z5[.Oj?nheGZ5YPdx4p defined & explained in the simplest way possible. << /S /GoTo /D [5 0 R /Fit] >> Published since 1866 continuously, Lehigh University course catalogs contain academic announcements, course descriptions, register of names of the instructors and administrators; information on buildings and grounds, and Lehigh history. Home / / probability of finding particle in classically forbidden region. Zoning Sacramento County, Give feedback. probability of finding particle in classically forbidden region The turning points are thus given by En - V = 0. For the particle to be found . \[\delta = \frac{1}{2\alpha}\], \[\delta = \frac{\hbar x}{\sqrt{8mc^2 (U-E)}}\], The penetration depth defines the approximate distance that a wavefunction extends into a forbidden region of a potential. Track your progress, build streaks, highlight & save important lessons and more! endobj probability of finding particle in classically forbidden region. what is jail like in ontario; kentucky probate laws no will; 12. /Subtype/Link/A<> 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 $|\psi(x, t)|^2$. Is it just hard experimentally or is it physically impossible? The probability of finding the particle in an interval x about the position x is equal to (x) 2 x. But for . How to notate a grace note at the start of a bar with lilypond? endobj (4.303). [1] J. L. Powell and B. Crasemann, Quantum Mechanics, Reading, MA: Addison-Wesley, 1961 p. 136. If you work out something that depends on the hydrogen electron doing this, for example, the polarizability of atomic hydrogen, you get the wrong answer if you truncate the probability distribution at 2a. The speed of the proton can be determined by relativity, \[ 60 \text{ MeV} =(\gamma -1)(938.3 \text{ MeV}\], \[v = 1.0 x 10^8 \text{ m/s}\] The integral you wrote is the probability of being betwwen $a$ and $b$, Sorry, I misunderstood the question. We've added a "Necessary cookies only" option to the cookie consent popup. classically forbidden region: Tunneling . However, the probability of finding the particle in this region is not zero but rather is given by: What is the probability of finding the partic 1 Crore+ students have signed up on EduRev. I don't think it would be possible to detect a particle in the barrier even in principle. 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. Classically this is forbidden as the nucleus is very strongly being held together by strong nuclear forces. /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 >> Particle Properties of Matter Chapter 14: 7. MathJax reference. You can see the sequence of plots of probability densities, the classical limits, and the tunneling probability for each . Do roots of these polynomials approach the negative of the Euler-Mascheroni constant? Slow down electron in zero gravity vacuum. Quantum tunneling through a barrier V E = T . endobj . 2. rev2023.3.3.43278. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. 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. The green U-shaped curve is the probability distribution for the classical oscillator. June 23, 2022 Solutions for What is the probability of finding the particle in classically forbidden region in ground state of simple harmonic oscillatorCorrect answer is '0.18'. Making statements based on opinion; back them up with references or personal experience. Connect and share knowledge within a single location that is structured and easy to search. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Probability of finding a particle in a region. 21 0 obj Is it possible to create a concave light? ncdu: What's going on with this second size column? In classically forbidden region the wave function runs towards positive or negative infinity. There are numerous applications of quantum tunnelling. 12 0 obj 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. a is a constant. Now if the classically forbidden region is of a finite width, and there is a classically allowed region on the other side (as there is in this system, for example), then a particle trapped in the first allowed region can . Peter, if a particle can be in a classically forbidden region (by your own admission) why can't we measure/detect it there? (iv) Provide an argument to show that for the region is classically forbidden. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Como Quitar El Olor A Humo De La Madera, This dis- FIGURE 41.15 The wave function in the classically forbidden region. See Answer please show step by step solution with explanation 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. The same applies to quantum tunneling. If so, how close was it? For simplicity, choose units so that these constants are both 1. Thus, the energy levels are equally spaced starting with the zero-point energy hv0 (Fig. /Border[0 0 1]/H/I/C[0 1 1] Particle always bounces back if E < V . You don't need to take the integral : you are at a situation where $a=x$, $b=x+dx$. has been provided alongside types of What is the probability of finding the particle in classically forbidden region in ground state of simple harmonic oscillatorCorrect answer is '0.18'. << There is nothing special about the point a 2 = 0 corresponding to the "no-boundary proposal". Do you have a link to this video lecture? Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. 1996-01-01. In particular, it has suggested reconsidering basic concepts such as the existence of a world that is, at least to some extent, independent of the observer, the possibility of getting reliable and objective knowledge about it, and the possibility of taking (under appropriate . Has a particle ever been observed while tunneling? Classically, there is zero probability for the particle to penetrate beyond the turning points and . The best answers are voted up and rise to the top, Not the answer you're looking for? Classically forbidden / allowed region. Can I tell police to wait and call a lawyer when served with a search warrant? These regions are referred to as allowed regions because the kinetic energy of the particle (KE = E U) is a real, positive value. For the first few quantum energy levels, one . From: Encyclopedia of Condensed Matter Physics, 2005. (a) Show by direct substitution that the function, probability of finding particle in classically forbidden region. beyond the barrier. endobj (4), S (x) 2 dx is the probability density of observing a particle in the region x to x + dx. . and as a result I know it's not in a classically forbidden region? The integral in (4.298) can be evaluated only numerically. You've requested a page on a website (ftp.thewashingtoncountylibrary.com) that is on the Cloudflare network. This property of the wave function enables the quantum tunneling. >> ample number of questions to practice What is the probability of finding the particle in classically forbidden region in ground state of simple harmonic oscillatorCorrect answer is '0.18'. Seeing that ^2 in not nonzero inside classically prohibited regions, could we theoretically detect a particle in a classically prohibited region? quantum-mechanics 19 0 obj A scanning tunneling microscope is used to image atoms on the surface of an object. Whats the grammar of "For those whose stories they are"? So anyone who could give me a hint of what to do ? 2. Wavepacket may or may not . /D [5 0 R /XYZ 200.61 197.627 null] \int_{\sqrt{5} }^{\infty }(4y^{2}-2)^{2} e^{-y^{2}}dy=0.6740. Interact on desktop, mobile and cloud with the free WolframPlayer or other Wolfram Language products. endobj This page titled 6.7: Barrier Penetration and Tunneling is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Paul D'Alessandris. You simply cannot follow a particle's trajectory because quite frankly such a thing does not exist in Quantum Mechanics. Surly Straggler vs. other types of steel frames. We need to find the turning points where En. For a quantum oscillator, we can work out the probability that the particle is found outside the classical region. before the probability of finding the particle has decreased nearly to zero. << We have step-by-step solutions for your textbooks written by Bartleby experts! Energy and position are incompatible measurements. For a quantum oscillator, we can work out the probability that the particle is found outside the classical region. Particle always bounces back if E < V . E.4). 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 You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Published since 1866 continuously, Lehigh University course catalogs contain academic announcements, course descriptions, register of names of the instructors and administrators; information on buildings and grounds, and Lehigh history. If I pick an electron in the classically forbidden region and, My only question is *how*, in practice, you would actually measure the particle to have a position inside the barrier region. The turning points are thus given by En - V = 0. >> And since $\cos^2+\sin^2=1$ regardless of position and time, does that means the probability is always $A$? Euler: A baby on his lap, a cat on his back thats how he wrote his immortal works (origin? in the exponential fall-off regions) ? Powered by WOLFRAM TECHNOLOGIES ${{\int_{a}^{b}{\left| \psi \left( x,t \right) \right|}}^{2}}dx$. << Step 2: Explanation. A typical measure of the extent of an exponential function is the distance over which it drops to 1/e of its original value. ross university vet school housing. Classical Approach (Part - 2) - Probability, Math; Video | 09:06 min. This expression is nothing but the Bohr-Sommerfeld quantization rule (see, e.g., Landau and Lifshitz [1981]). 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). a) Locate the nodes of this wave function b) Determine the classical turning point for molecular hydrogen in the v 4state. Why is the probability of finding a particle in a quantum well greatest at its center? I think I am doing something wrong but I know what! << calculate the probability of nding the electron in this region. . "`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 ,i V _"QQ xa0=0Zv-JH 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. Misterio Quartz With White Cabinets, Hmmm, why does that imply that I don't have to do the integral ? We can define a parameter defined as the distance into the Classically the analogue is an evanescent wave in the case of total internal reflection. so the probability can be written as 1 a a j 0(x;t)j2 dx= 1 erf r m! Wave vs. We know that for hydrogen atom En = me 4 2(4pe0)2h2n2. Confusion regarding the finite square well for a negative potential. This is impossible as particles are quantum objects they do not have the well defined trajectories we are used to from Classical Mechanics. Contributed by: Arkadiusz Jadczyk(January 2015) It is easy to see that a wave function of the type w = a cos (2 d A ) x fa2 zyxwvut 4 Principles of Photoelectric Conversion solves Equation (4-5). Why Do Dispensaries Scan Id Nevada, 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. Does a summoned creature play immediately after being summoned by a ready action? /Type /Annot In the ground state, we have 0(x)= m! endobj "After the incident", I started to be more careful not to trip over things. The values of r for which V(r)= e 2 . This is . 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. When the tip is sufficiently close to the surface, electrons sometimes tunnel through from the surface to the conducting tip creating a measurable current. Question about interpreting probabilities in QM, Hawking Radiation from the WKB Approximation. Forbidden Region. /ProcSet [ /PDF /Text ] 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. They have a certain characteristic spring constant and a mass. +2qw-\ \_w"P)Wa:tNUutkS6DXq}a:jk cv 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 . I'm supposed to give the expression by $P(x,t)$, but not explicitly calculated. 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. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. zero probability of nding the particle in a region that is classically forbidden, a region where the the total energy is less than the potential energy so that the kinetic energy is negative. The same applies to quantum tunneling. (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 . theory, EduRev gives you an What is the point of Thrower's Bandolier? /Annots [ 6 0 R 7 0 R 8 0 R ] First, notice that the probability of tunneling out of the well is exactly equal to the probability of tunneling in, since all of the parameters of the barrier are exactly the same. endstream in English & in Hindi are available as part of our courses for Physics. If so, why do we always detect it after tunneling. H_{4}(y)=16y^{4}-48y^{2}-12y+12, H_{5}(y)=32y^{5}-160y^{3}+120y. The probability is stationary, it does not change with time. Also assume that the time scale is chosen so that the period is . Therefore the lifetime of the state is: 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. quantum mechanics; jee; jee mains; Share It On Facebook Twitter Email . /D [5 0 R /XYZ 234.09 432.207 null] Textbook solution for Introduction To Quantum Mechanics 3rd Edition Griffiths Chapter 2.3 Problem 2.14P. 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. Legal. In general, we will also need a propagation factors for forbidden regions. Go through the barrier . In general, quantum mechanics is relevant when the de Broglie wavelength of the principle in question (h/p) is greater than the characteristic Size of the system (d). we will approximate it by a rectangular barrier: The tunneling probability into the well was calculated above and found to be 2 More of the solution Just in case you want to see more, I'll . Energy eigenstates are therefore called stationary states . /D [5 0 R /XYZ 126.672 675.95 null] Has a double-slit experiment with detectors at each slit actually been done? But there's still the whole thing about whether or not we can measure a particle inside the barrier. Can you explain this answer? Finding particles in the classically forbidden regions [duplicate]. The answer is unfortunately no. 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. The Franz-Keldysh effect is a measurable (observable?) That's interesting. (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 )}. We know that for hydrogen atom En = me 4 2(4pe0)2h2n2. xZrH+070}dHLw sage steele husband jonathan bailey ng nhp/ ng k . 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. Qfe lG+,@#SSRt!(` 9[bk&TczF4^//;SF1-R;U^SN42gYowo>urUe\?_LiQ]nZh The zero-centered form for an acceptable wave function for a forbidden region extending in the region x; SPMgt ;0 is where . >> /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R \int_{\sqrt{3} }^{\infty }y^{2}e^{-y^{2}}dy=0.0495. I'm not really happy with some of the answers here. for 0 x L and zero otherwise. Classically the particle always has a positive kinetic energy: Here the particle can only move between the turning points and , which are determined by the total energy (horizontal line). Particle in a box: Finding <T> of an electron given a wave function. Consider the hydrogen atom. 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 symmetry, the probability of the particle being found in the classically forbidden region from x_{tp} to is the same. 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. Is a PhD visitor considered as a visiting scholar? \int_{\sqrt{7} }^{\infty }(8y^{3}-12y)^{2}e^{-y^{2}}dy=3.6363. 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. Calculate the radius R inside which the probability for finding the electron in the ground state of hydrogen . Title . He killed by foot on simplifying. 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 $E_ {3}$. A particle in an infinitely deep square well has a wave function given by ( ) = L x L x 2 2 sin. The wave function oscillates in the classically allowed region (blue) between and . find the particle in the . 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). Although it presents the main ideas of quantum theory essentially in nonmathematical terms, it . (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. The best answers are voted up and rise to the top, Not the answer you're looking for? [3] How to match a specific column position till the end of line? If we can determine the number of seconds between collisions, the product of this number and the inverse of T should be the lifetime () of the state: Which of the following is true about a quantum harmonic oscillator? Is there a physical interpretation of this? I am not sure you could even describe it as being a particle when it's inside the barrier, the wavefunction is evanescent (decaying). \[ \delta = \frac{\hbar c}{\sqrt{8mc^2(U-E)}}\], \[\delta = \frac{197.3 \text{ MeVfm} }{\sqrt{8(938 \text{ MeV}}}(20 \text{ MeV -10 MeV})\]. Year . ~! << Also, note that there is appreciable probability that the particle can be found outside the range , where classically it is strictly forbidden! Such behavior is strictly forbidden in classical mechanics, according to which a particle of energy is restricted to regions of space where (Fitzpatrick 2012). /Type /Annot L2 : Classical Approach - Probability , Maths, Class 10; Video | 09:06 min. In metal to metal tunneling electrons strike the tunnel barrier of height 3 eV from SE 301 at IIT Kanpur /Border[0 0 1]/H/I/C[0 1 1] This Demonstration shows coordinate-space probability distributions for quantized energy states of the harmonic oscillator, scaled such that the classical turning points are always at . Gloucester City News Crime Report, \[ \tau = \bigg( \frac{15 x 10^{-15} \text{ m}}{1.0 x 10^8 \text{ m/s}}\bigg)\bigg( \frac{1}{0.97 x 10^{-3}} \]. The difference between the phonemes /p/ and /b/ in Japanese, Difficulties with estimation of epsilon-delta limit proof. To me, this would seem to imply negative kinetic energy (and hence imaginary momentum), if we accept that total energy = kinetic energy + potential energy. /D [5 0 R /XYZ 261.164 372.8 null] Belousov and Yu.E. We will have more to say about this later when we discuss quantum mechanical tunneling. Probability of particle being in the classically forbidden region for the simple harmonic oscillator: a. This Demonstration calculates these tunneling probabilities for . What is the probability of finding the particle in classically forbidden region in ground state of simple harmonic oscillator. Non-zero probability to . The classical turning points are defined by E_{n} =V(x_{n} ) or by \hbar \omega (n+\frac{1}{2} )=\frac{1}{2}m\omega ^{2} x^{2}_{n}; that is, x_{n}=\pm \sqrt{\hbar /(m \omega )} \sqrt{2n+1}. In particular the square of the wavefunction tells you the probability of finding the particle as a function of position. (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. /Subtype/Link/A<> When the width L of the barrier is infinite and its height is finite, a part of the wave packet representing . << Mathematically this leads to an exponential decay of the probability of finding the particle in the classically forbidden region, i.e. Thus, the particle can penetrate into the forbidden region. Reuse & Permissions In fact, in the case of the ground state (i.e., the lowest energy symmetric state) it is possible to demonstrate that the probability of a measurement finding the particle outside the . Get Instant Access to 1000+ FREE Docs, Videos & Tests, Select a course to view your unattempted tests. 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. 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'. This is referred to as a forbidden region since the kinetic energy is negative, which is forbidden in classical physics. E is the energy state of the wavefunction. Are these results compatible with their classical counterparts? (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.). Probability for harmonic oscillator outside the classical region, We've added a "Necessary cookies only" option to the cookie consent popup, Showing that the probability density of a linear harmonic oscillator is periodic, Quantum harmonic oscillator in thermodynamics, Quantum Harmonic Oscillator Virial theorem is not holding, Probability Distribution of a Coherent Harmonic Oscillator, Quantum Harmonic Oscillator eigenfunction.
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probability of finding particle in classically forbidden region