Now let's define for convenience the parameters \( \xi \equiv ka \) and \( \eta \equiv \kappa a \). Similarly, for the odd-parity states we find Which can be combined to obtain a condition on the wave numbers and thus the allowed energy \( E \): Recall that the most general even-parity solution takes the formĪpplying the boundary condition at \( x=a \) (the other boundary is redundant) to this state gives We had just appealed to parity symmetry to separate the energy eigenfunction solutions into even and odd parity. As a consequence, there are even states in which ψ e(− x) = ψ e( x), and odd states in which ψ o(− x) = −ψ o( x).At the end of last time, we were finishing up our solution for bound states in the square well. We note that as the potential energy well gets shallower and/or narrower, ζ 0 0.ĢSince E < 0, we choose to write E = −| E| to avoid any ambiguity in sign.ģThis is due to the fact that for even potential energy functions, the Hamiltonian commutes with the parity operator. You can change a and |V 0| by dragging the sliders to a particular value to see how the left-hand side of Eq. You may also select the Show the transcendental equation as a function of energy instead link to see the equations as a function of energy. (11.7) is shown in black and the left-hand side is shown in red. We can solve this equation numerically or graphically, and we choose graphically in the animation. Energy levels in quantum wells with capping barrier layer of finite size: Bound states and oscillatory behavior of the continuum states. (11.7) only has solutions for particular values of ζ. This equation is a transcendental equation for ζ which itself is related via Eq. General series solution for finite square-well energy levels for use in wave-packet studies. We now consider the following substitutions in terms of dimensionless variables: This is actually a constraint on the allowed energies, as both k and κ involve the energy. We now divide the resulting two equations to give a condition for the existence of even solutions: κ/ k = tan( ka). Ψ' I(− a) = ψ' II(− a) → Aκ exp(−κ a) = Cksin(− ka).įrom the symmetry in the problem, we need not consider the boundary at x = a as it yields the exact same condition on energy eigenfunctions. Matching proceeds much like the scattering cases we considered in Chapter 8. 3 We begin by considering the even (parity) solutions and therefore the ψ II(x) = Ccos( kx) solution in Region II. Since the potential energy function is symmetric about the origin, there are even and odd parity solutions to the bound-state problem. In his classical textbook The Universe in a Helium Droplet, Volovik described an interesting autonomously isolated quantum system of helium nanodroplet, without any interaction with the surrounding environment.It can be in equilibrium at zero external pressure (i.e. Related Threads on Energies and numbers of bound states in finite potential well I Finite square well bound states. Matching the energy eigenfunctions across these boundaries means that the energy eigenfunctions and the first derivatives of the energy eigenfunctions must match at each boundary so that we have a continuous and smooth wave function (no jumps or kinks). Energies and numbers of bound states in finite potential well Thread starter 71GA Start date Apr 5, 2013. Next, we must match the solutions across the boundaries at x = − a and x = a. The above equation has the solutions ψ II( x) = Bsin( kx) Ccos( kx) which are valid solutions for Region II. In this region we can write the time-independent Schrödinger as V( x) = −| V 0| − a a and x a), for bound-state solutions, E V 0 or | V 0| > | E|. The finite square well problem is defined by a potential energy function that is zero everywhere except 1 Please wait for the animation to completely load. Restart | Show the transcendental equation as a function of energy instead | Show the energy eigenfunction and well instead Check, then drag a slider to see the odd states' TE.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |