{"history":[],"article":{"id":"bd835ef1-d15f-4186-b240-b37bac36e66a","title":"Finite potential well","slug":"finite-potential-well","content":":::summary\n:::\n\n# \nThe **finite potential well** (also known as the **finite square well**) is a concept from [quantum mechanics](https://bliptext.com/articles/quantum-mechanics).\nIt is an extension of the [infinite potential well](https://bliptext.com/articles/infinite-[potential](https://bliptext.com/articles/potential-energy)-well), in which a particle is confined to a \"box\", but one which has finite [potential](https://bliptext.com/articles/potential-energy) \"walls\".\nUnlike the infinite potential well, there is a [probability](https://bliptext.com/articles/probability) associated with the particle being found outside the box.\nThe quantum mechanical interpretation is unlike the classical interpretation, where if the total [energy](https://bliptext.com/articles/energy) of the particle is less than the potential [energy](https://bliptext.com/articles/energy) barrier of the walls it cannot be found outside the box.\nIn the quantum interpretation, there is a non-zero probability of the particle being outside the box even when the energy of the particle is less than the potential energy barrier of the walls (cf [quantum tunnelling](https://bliptext.com/articles/quantum-tunnelling)).\n\n# Particle in a one-dimensional potential well\nFor the one-dimensional case on the x-axis, the [time-independent Schrödinger equation](https://bliptext.com/articles/time-independent-schr-dinger-equation) can be written as:\nwhere\nFor the case of the particle in a one-dimensional box of length L, the potential is V_0 outside the box, and zero for x between -L/2 and L/2.\nThe wavefunction is composed of different wavefunctions; depending on whether x is inside or outside of the box, such that:\n\n## Inside the box\nFor the region inside the box, V(x) = 0 and Equation 1 reduces to\nresembling the time-independent [free schrödinger equation](https://bliptext.com/articles/free-particle), hence\nLetting\nthe equation becomes\nwith a general solution of\nwhere A and B can be any [complex numbers](https://bliptext.com/articles/complex-number), and k can be any real number.\n\n## Outside the box\nFor the region outside of the box, since the potential is constant, V(x) = V_0 and equation becomes:\nThere are two possible families of solutions, depending on whether E is less than V_0 (the particle is in a bound state) or E is greater than V_0 (the particle is in an unbounded state).\nIf we solve the time-independent Schrödinger equation for an energy E > V_0, letting such that\nthen the solution has the same form as the inside-well case:\nand, hence, will be oscillatory both inside and outside the well.\nThus, the solution is never square integrable; that is, it is always a non-normalizable state.\nThis does not mean, however, that it is impossible for a quantum particle to have energy greater than V_0, it merely means that the system has continuous spectrum above V_0, i.e., the non-normalizable states still contribute to the continuous part of the spectrum as [generalized eigenfunctions](https://bliptext.com/articles/dirac-delta-function) of an unbounded operator.\nThis analysis will focus on the bound state, where E < V_0.\nLetting\nproduces\nwhere the general solution is exponential:\nSimilarly, for the other region outside the box:\nNow in order to find the specific solution for the problem at hand, we must specify the appropriate boundary conditions and find the values for A, B, F, G, H and I that satisfy those conditions.\n\n## Finding wavefunctions for the bound state\nSolutions to the Schrödinger equation must be continuous, and continuously differentiable.\nThese requirements are [boundary conditions](https://bliptext.com/articles/boundary-condition) on the differential equations previously derived, that is, the matching conditions between the solutions inside and outside the well.\nIn this case, the finite potential well is symmetrical, so symmetry can be exploited to reduce the necessary calculations.\nSummarizing the previous sections:\nwhere we found \\psi_1, \\psi_2, and \\psi_3 to be:\nWe see that as x goes to -\\infty, the F term goes to infinity.\nLikewise, as x goes to +\\infty, the I term goes to infinity.\nIn order for the wave function to be square integrable, we must set F = I = 0, and we have:\nand\nNext, we know that the overall \\psi function must be continuous and differentiable.\nIn other words, the values of the functions and their derivatives must match up at the dividing points:\nThese equations have two sorts of solutions, symmetric, for which A = 0 and G = H, and antisymmetric, for which B = 0 and G=-H.\nFor the symmetric case we get\nso taking the ratio gives\nSimilarly for the antisymmetric case we get\nRecall that both \\alpha and k depend on the energy.\nWhat we have found is that the continuity conditions cannot be satisfied for an arbitrary value of the energy; because that is a result of the infinite potential well case.\nThus, only certain energy values, which are solutions to one or either of these two equations, are allowed.\nHence we find that the energy levels of the system below V_0 are discrete; the corresponding eigenfunctions are [bound states](https://bliptext.com/articles/bound-state).\n(By contrast, for the energy levels above V_0 are continuous. )\nThe energy equations cannot be solved analytically.\nNevertheless, we will see that in the symmetric case, there always exists at least one bound state, even if the well is very shallow.\nGraphical or numerical solutions to the energy equations are aided by rewriting them a little and it should be mentioned that a nice approximation method has been found by Lima which works for any pair of parameters L and V_0.\nIf we introduce the dimensionless variables and v=k L/2, and note from the definitions of \\alpha and k that, where , the master equations read\nIn the plot to the right, for u_0^2=20, solutions exist where the blue semicircle intersects the purple or grey curves (v \\tan v and -v \\cot v).\nEach purple or grey curve represents a possible solution, v_j within the range.\nThe total number of solutions, N, (i.e., the number of purple/grey curves that are intersected by the blue circle) is therefore determined by dividing the radius of the blue circle, u_0, by the range of each solution \\pi/2 and using the floor or ceiling functions:\nIn this case there are exactly three solutions, since.\nand v_3=3.73, with the corresponding energies\nIf we want, we can go back and find the values of the constants A, B, G, H in the equations now (we also need to impose the normalisation condition).\nOn the right we show the energy levels and wave functions in this case (where ).\nWe note that however small u_0 is (however shallow or narrow the well), there is always at least one bound state.\nTwo special cases are worth noting.\nAs the height of the potential becomes large,, the radius of the semicircle gets larger and the roots get closer and closer to the values v_n=n\\pi/2, and we recover the case of the [infinite square well](https://bliptext.com/articles/infinite-square-well).\nThe other case is that of a very narrow, deep well - specifically the case and L\\to 0 with V_0 L fixed.\nAs it will tend to zero, and so there will only be one bound state.\nThe approximate solution is then, and the energy tends to.\nBut this is just the energy of the bound state of a [Delta function potential](https://bliptext.com/articles/delta-function-potential) of strength V_0 L, as it should be.\nA simpler graphical solution for the energy levels can be obtained by normalizing the potential and the energy through multiplication by.\nThe normalized quantities are\ngiving directly the relation between the allowed couples (V_0, E) as\nfor the even and odd parity wave functions, respectively.\nIn the previous equations only the positive derivative parts of the functions have to be considered.\nThe chart giving directly the allowed couples (V_0, E) is reported in the figure.\n# [finite potential well | upload.wikimedia.org/wikipedia/commons/2/26/FigureV0E///QuantumWell.png]\n\n## Asymmetric well\nConsider a one-dimensional asymmetric potential well given by the potential\nwith V_2 > V_1.\nThe corresponding solution for the wave function with E V_0, letting such that\nthen the solution has the same form as the inside-well case:\nand, hence, will be oscillatory both inside and outside the well.\nThus, the solution is never square integrable; that is, it is always a non-normalizable state.\nThis does not mean, however, that it is impossible for a quantum particle to have energy greater than V_0, it merely means that the system has continuous spectrum above V_0, i.e., the non-normalizable states still contribute to the continuous part of the spectrum as [generalized eigenfunctions](https://bliptext.com/articles/dirac-delta-function) of an unbounded operator.\nThis analysis will focus on the bound state, where E V_1.\nThe corresponding solution for the wave function with E

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