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Particle in a Box Particle in a Box V(x) = V(x) = I II III V(x) = 0 The QM postulates insist that (x) be continuous and single valued. We will solve the Sch. Eqnseppygarately in regions I, II and III and then put the pieces together to obey the above constraints. n22π 2 nh 22 E where n 1231,2,3, … n 2ma22 8ma nx nnx A sin for 0xa, and 0 othitherwise a ConcepTest #1 Consider the energy difference between the n=1 and n=2 levels for a marble of mass 1 g confined in a one-dimensional box of length 0.10 m. Consider the wavelength that corresponds to a spectral transition between these levels. How does E between n=1 and 2 for the marble compare to that for the e– in the previous ( 4Å, 1D box) problem? How does for this spectral transition for the marble compare to that for the e– in the previous problem ? A. larger, larger D. smaller, smaller B. larger, smaller E. smaller, larger C. same, same We now need to see what happens as we add more degrees of freedom to a quantum system. Two independent 1-D particles in two boxes Consider two ppparticles in two separate infinite 1-D wells. Their wave functions are zero outside their own box, so they cannot possibly know about each other. M1 widths a and b M2 a b From earlier, the energy of each must be quantized . Also, the total energy must be the sum of the iidindiv idua l energies. n222 n222 1 2 Etotal 22 n 1 1,2,3 n 2 1,2,3 22bMa12 M Notice that there are now two quantum numbers, associated with the two degrees of freedom with constrained motion. We will see that the two particle wave function is the product of the individual wave functions. Let’s actually solve this problem Two independent 1-D particles in two boxes Since the two particles do not communicate, I can write their coordinates as x1 and x2, and express the potential potential energy in the form V xx12, 0 for 0xa 1 and 0xb 2 for x11 < 0 or x > a for x22 < 0 or x > b This shows that we can write V xx12, Vab x 1 V x 2 WiiWriting the Hamil toni an in terms of this potentiilal, 22 22 ˆ xx12, 22 Vab x 1 V x 2 22MM12xx12 22 22 Vx Vx hx hx 22MM22abab1212 12xx12 We have set the stage to separate the Schrödinger equation with two variables into two separate equations, each with one variable. Two independent 1-D particles in two boxes We assume a solution of the form xx12, fx 1 gx 2 ,a product form and then can show it to be a general so luti on The Schrödinger equation is hx h x fx gx gx hxfx fxh x gx Efx gx ab 1 2 12 2 a 11 1 b 22 12 Dividing by f(x1)g(x2) and rearranging a bit, hf x h g x ab12E fx12 gx This can only be true if each side is a constant. Why?? Let l.h.s. = C Then hfa x11 Cf x , the particle in a (0,a) box Sch equation Similarly, we get hgb x22 Kgx , the particle in a (0,b) box Sch equation Two independent 1-D particles in two boxes The result is that the total energy is the sum of the separately quantized particle in a box energies, and The two particle wave function is the product of the two individual wave functions. Two degrees of freedom with constrained motion: Two quantum numbers! This result generalizes and is critically important. Separable!! Additional Dimensions/Particles M1 M2 Two independent 1-D particles in two boxes widths a and b 22 22 ˆ xx12, 22 Vx 1 Vx 2 22MM12xx12 Separable Two independent partclesparticles in two boxes n222 n222 1 2 Etotal 22 n 1 1,2,3 n 2 1,2,3 22bMa12 M 22 nx x Two quantum numbers 11n 2 2 nn xx12, sin sin 12 ab a b for 0 xaand 0 y b Separability of Solutions If the multi coordinate Hamiltonian operator has the special form (qqq123 , , q n) =hh12 ( q 1 ) + ( q 2 ) + h3 ( q 3 )+ + hn ( q n ) with each component having the form hqii Tq ii Vq ii then there exists a simple, general solution n (,qq , , q ) q q q 12 n 12()()()123 3 i ()qi i=1 with each k satisfying a 1D Schrödinger-like equation hEkk kk. n and the total energy is the sum of individual components, EE i . i 1 Thus whenever the Hamiltonian can be written as a sum, then becomes a product of one variable terms and E becomes a sum of energies associated with each degree of freedom. The difficulty of obtaining a separable solution scales linearly with the number of degrees of freedom, and any 1D problem is very easy! We will do (almost) anything to achieve separability! Additional Dimensions/Particles Two non-interacting 1-D particles in a box of width a 22 22 ˆ xx12, 22 Vx 22mm12xx12 2 Separable, since we neglected gravitational attraction, (m1m2)/(x2-x1) n222 n222 1 2 Etotal 22 n 1 1,2,3 n 2 1,2,3 22mma12 a Two quantum numbers 22 nx x 11n 2 2 nn xx12, sin sin 12 aa a a for 0 x12aand 0 x a Mass m particle in a two-dimensional (x,y) box V(x, y) = , for x < 0 or x > a or for y < 0 or y > b, and a = 0 , for 0 x a and 0 y b m 2 2 b p py TTVxy,, x Vxy xy 22mm 22 22 ˆ xy, 0 inside the rectangle (separable!) 22 22mmxy 2 2 2 2 ()y d ()x d ()x - 2 E y ()y 2 E x 2m 2m dx dy 2 2 2 2 2 2 n1 n2 E n 1,2,3, Ey n 1,2,3, x 2ma 2 1 2mb2 2 2 nx1 2 ny2 (x) sin n(y) sin n1 aa bb for 0 x aand 0 otherwise for 0 y b and 0 otherwise 22 nx1 n2 y nn x, y sin sin 12 ab a b f0for 0 x aandbd 0 y b n222 n222 E 1 2 n 1,2,3 n 1,2,3 total22mbma22 1 2 Particle in a two-dimensional (x,y) box a 22 nx y x, y sin1 sinn2 m nn b 12 ab a b for 0 xaand 0 y b n222 n222 E 1 2 n 1,2,3 n 1,2,3 total22mbma 22 1 2 Note especially!! There are two degrees of freedom with constrained motion and the result is that there are two quantum numbers, n1 and n2. What do the solutions look like? Nodal line ψ=0 (2D) (points in 1D) (surface in 3D) 1,2,3 for Particle in a 1D Box In one dimension, the nodes of the standing deBroglie waves are points. The nodal points become lines in 2D, and surfaces in 3D. The energy levels in 1D are rigorously ordered by the number of nodes. This is not quite the case in higher dimensions, as we will ss.ee. Particle in a two-dimensional (x,y) box Consider the 2,1 state as well. Now what is the total energy of the 2,1 state? It is the same as the 1,2 state. Particle in a two-dimensional (x,y) box The two state functions correspond to the same energy, but they are very different spatially. This is called degeneracy! Here, twofold degenerate. In fact, these two functions are orthogonal!.
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