Band Structure and matrix Methods

Transcription

1 Quantum Mechanics Physics 34 -Winter 0-University of Chicago Outline Band Structure and matrix Methods Jing Zhou ID:4473 March 0, 0 Introduction Supersymmetric Quantum Mechanics and Inverse Scattering 3 Matrix Methods for One-dimensional Problems 4 Some Examples of Band Structure 5 Conclusion Introduction It has been found useful in practical applications to ignore scattering data and to construct potentials from bound state data alone, i.e, in the study of quark-antiquark bound states. In this paper, we talk about supersymmetric quantum mechanics and inverse scattering. We use supersymmetric quantum mechanics to reconstruct potential from known band structure. Second, matrix methods for one-dimensional methods are presented in this paper. At the same time, in order to understand this method better, two examples are given. One is a periodic delta function. The other is a double delta comb. Using matrix methods, we get the energy band. The band structure pictures are presented. Supersymmetric Quantum Mechanics and Inverse Scattering We begin by reviewing the factorization method for the hamiltonian H d dx V x, where V x is a given potential. Writing H A A, with A d dx x and

2 A d dx x then we find x must satisfy V x 3 4 The differentiation is with respect to the position x. The eigenfunctions ψ of H for negative energy E κ satisfy the equation H ψ A Aψ E ψ 5 If we multiply this equation by the operator A, AH ψ AA Aψ E Aψ 6 we find the function Aψ ψ satisfies an eigenvalue equation with the same eigenvalue E but a different operator H AA H ψ E ψ 7 unless Aψ 0, in which case ψ is the ground-state eigenfunction of H. In this case the spectrum of H contains one more bound state than that of H. This additional bound state must lie below any of the others. If we define the potential V x H d dx V x Since H AA, it implies V x 8 The above method allows us to start with a potential V which is zero and relate it to a potential V which has a single negative energy bound state at E κ. We simply solve the equation V κ κ for x κ tanh κ x x, where x is an integration constant, and then we find from V κ thatv x κ sech κ x x. This is the form of a solition of the KdV equation at any instant of time. It is nature to continue the work to get x and then V x till x and V x. Potentials with bound states at energies E κ, n,,n may be constructed by solving the sequence of equations V κ 9a V κ 9b Eqs. (9) can be expressed in terms of a partial differential equation if we assume that κ is slowly varying with n. Let n. Define ε κ. Then (9b) can be written as V ε 0

3 Then from (9b) -(9a), we get which is V V V x Then we differentiate Eq. (0) with respect to y and substitute for V, y x y dε x dy Now, if we know the energy band of ε, we can get from solving Eq. (), then get the potential. Now let us consider a simple case. Suppose we the know the energy band ε cy which is equally spaced with c a constant. If we try a solution for a polynomial is x and y, we find that, where is another constant. Then we obtain the potential: V x, y c x x c y We are exciting that this is a family of harmonic oscillator potentials as we expected. Figure : energy levels of harmonic oscillator 3 Matrix Method for One-dimensional problems If we have a potential, and we want to know the wave function and its corresponding energy levels, usually we can solve the Schrodinger equation to get what we want. and So d H p V V x m m dx d ψ x dx Hψ x Eψ x E V x m 0 3

4 For the solution of many one-dimensional quantum mechanics problems, it would be convenient to rewrite the Schrodinger equation for the wave function ψ x in a potential V x with energy E in matrix form: where Ψ ΜΨ 3, Ψ ψ ψ, 4 and 0 Μ 5 V E 0 For a constant potential V, we can solve Eq. 3 dψ Μdx Ψ lnψ Mx Ψ exp Mx So Ψ x Δ exp ΜΔ Ψ x T Δ Ψ x For V E 0, we find that the T Δ ( the "transfer matrix") is given by T Δ coshλδ λ sinhλδ 6a λsinhλδ coshλδ For E V 0, the corresponding form is T Δ cos Δ k sin Δ 6b sin Δ cos Δ Now let us talk more about the properties of this transfer matrix. ) It is easy to find that det T Δ, and T Δ T Δ. ) The matrix that propagates a particle through a sequence of piecewise constant potentials is the product of the matrices for each region. 3) The matrix(6b) propagates a free particle with energy E and wave function ψ x exp, so exp Ψ x exp exp Ψ x a exp exp T Ψ x then T exp 4) Consider a bound state with E 0, and let V 0 out side the region X, 0 x a. For a normalizable wave function, we must have 4

5 Ψ~ κ at x 0; and Ψ~ at x a κ Since we may demand T Ψ 0 Ψ a κ T 0 7 κ This provides an eigenvalue condition which can only be satisfied for certain E. 5) Suppose V αδ x x. By integrating the corresponding Schrodinger equation, we can get ψ x ε ψ x ε αψ x Ψ x ε ψ x ε ψ x ε T εψ x T ε ψ x ε ψ x ε andψ x ε ψ x ε ψ x, so ψ x Ψ x ε ψ x ε T T ψ x T T ψ x ε we can calculate T, get the corresponding matrix T α;δ 0 α When combined with Eq. (7), κ 0 α κ 0 this condition shows that an attractive delt-function potential has a single bound state at κ α. 6) Suppose a potential is periodic with period a. Then the eigenfunctions of the corresponding Hamiltonian will be Bloch waves, obeying ψ x a exp ψ x (More information in Ref []). Let the corresponding transfer matrix T be that which translates a solution by the amount a. Such a transfer matrix may be composed of the product of many elements; it obviously exists even in the case of a continuously varying potential. Its eigenvalues will be exp, since it must have unit determinant. Then the eigenvalue condition takes the form cosqa Tr 8 Eq. (8) could be proved below det T Δ T 0 with Δ exp T T Δ So T Δ T Δ T T 0; remembering that det T Δ, we get Δ T T Δ 0 that is Δ T T Δ Δ T T Δ where Δ exp, Δ exp ; doing some algebra, we can get Eq. (8). 5

6 The solutions of this condition will, in general, consist of bands. To well understand this method, two examples are presented in the next chapter. 4 Some Examples of Band Structures A Single delta function A periodic sequence of delta functions is one of the simplest one-dimensional systems which leads to a band structure. Let V α δ x n The choice of origin allows us to have V 0 0 and V x V x. First we consider the attractive situation with α 0. We seek solutions for bound states with energy E 0. The eigenvalue condition Eq. (8) reduces to cosq coshκ α sinhκ 9 κ Then we can use Matlab to draw the solution to this equation, shown in Figure. Figure : band structure for single attractive delta function with α,,4,6,8 For very small α 4, Eq. (9) becomes (using the corresponding expansion) α From Fig., We can see from the graph when α is or, the shape looks a ring with radius approximately equal to α. For α above the critical value of 4 ( the red line), energy gap develops, and the allowed values of k become confined to a band separated from zero. As the potential became very strong, the band becomes narrowler ( the range that k could reach is 6

7 smaller) confined to the region around κ α which is the value for a single delta-function. The wave functions are more and more localized around the delta functions. An approximation valid for large α is κ α e α cos q Second, let us look at the repulsive case in which α 0. The eigenvalue condition becomes cosq cosκ α κ sinκ The corresponding energy band is shown below in Figures 3 and 4. Band Structure of replusive delta function 7 =4 6 5 Band Structure of replusive delta function 7 = = 6 5 =4 =6 =8 4 4 k k q/ q/ Figure 3: band structure of repulsive Figure4: band structure of repulsive delta function with α 4 delta function with α,,4,6,8 B Double Delta Comb Now it is time to see the double delta comb case with the potential V α δ x n β δ x n The transfer matrix whose trace is cosq (note the periodicity in x is now ) is then the product of four matrices, two describing propagation of a particle over unit distance and two describing the delta function interactions. Again, first consider the attractive case of both α 0 β 0. The eigenvalue condition is then: 7

8 cos q coshκ α β sinhκ coshκ κ κ sinhκ The solutions are shown in Figures 5 and 6. 0 Band Structure of attractive double comb =3, =.8.6 =3, =.5 =3, =3.4. k q/ Figure 5: band structure of attractive double Figure 6: band structure of attractive double delta comb with β and α,.5, delta comb with β 3 and α,.5,3 Band Structure of attractive double comb 3.5 =6, = =6, = =6, =3 =6, =4 =6, =6 k q/ Figure 7: band structure of attractive double Figure 8: band structure of attractive double delta comb with β 4 and α,3, 4 delta comb with β 6 and α,.5,3, 4,6 For α and β, Eq. (0) becomes 8

9 α β For α and α β, the value 0 always corresponds to, as shown in Figure 5. For β 4, a two band structure can develop as long as α. And two more bands merge into one asα β. The band corresponding to smaller always contains the value 0. For large α and β, the eigenvalue condition are κ α 4 β α e cos q κ β 4 α β e cos q Then if both α 0 and β 0, which is the repulsive case, we have cos q cosκ α β sinκ cosκ κ κ sinκ The solutions are shown in Figures 9 and 0. Figure 9: band structure of repulsive double Figure 0: band structure of repulsive double delta comb with β and α,.5, delta comb with β 6 and α,.5,3, 4,6 9

10 5 Conclusion In this paper, we talk about how to use supersymmetric quantum mechanics to reconstruct potentials from known band structure. A matrix method to deal with one-dimensional quantum mechanics problems is presented. Two examples are given. One is a periodic delta function. The other is a double delta comb. The band structure pictures are presented. In fact, more work can be done. But due to the time limitation, I only did this. Future work should be expected. 6 Reference []J.L. Rosner, Ann. Phys. (NY)00, 0(990) [] Solid State Physics, Neil W. Ashcroft, N. David Mermin 0