方势垒定态波函数程序（Matlab）

                     

贡献者： addis

　　 这里给出计算方势垒定态波函数的 Matlab 程序

% m 质量，V0 势垒，E 能量， L 势垒半宽， odd 奇函数 true 偶函数 false
% A 中间区间系数， C 右边区间 cos 系数， D 右边区间 sin 系数

function [psi,A,C,D] = FSB(x, E,L,m,V0,odd)
k = sqrt(2*m*E);
psi = zeros(size(x));
m1 = (x < -L); m2 = (abs(x) <= L); m3 = (x > L);
if (E == V0), E = V0 + eps; end

if E >= V0
    b = sqrt(2*m*(E - V0));
    if ~odd
        A1 = 1/sqrt(pi*(cos(b*L)^2+b^2*sin(b*L)^2/k^2));
        C1 = A1*(cos(k*L)*cos(b*L) + b/k*sin(k*L)*sin(b*L));
        D1 = A1*(-b/k*cos(k*L)*sin(b*L) + sin(k*L)*cos(b*L));
        A = A1; C = C1; D = D1;
        psi(m1) = C1*cos(k*x(m1)) - D1*sin(k*x(m1));
        psi(m2) = A1*cos(b*x(m2));
        psi(m3) = C1*cos(k*x(m3)) + D1*sin(k*x(m3));
    else
        B2 = 1/sqrt(pi*(sin(b*L)^2+b^2*cos(b*L)^2/k^2));
        C2 = B2*(cos(k*L)*sin(b*L) - b/k*sin(k*L)*cos(b*L));
        D2 = B2*(b/k*cos(k*L)*cos(b*L) + sin(k*L) * sin(b*L));
        A = B2; C = C2; D = D2;
        psi(m1) =-C2*cos(k*x(m1)) + D2*sin(k*x(m1));
        psi(m2) = B2*sin(b*x(m2));
        psi(m3) = C2*cos(k*x(m3)) + D2*sin(k*x(m3));
    end
else
    ka = sqrt(2*m*(V0-E));
    if ~odd
        A1 = 1/sqrt(pi*(cosh(ka*L)^2+(ka/k)^2*sinh(k*L)^2));
        C1 = A1*(cosh(ka*L)*cos(k*L) - ka/k*sinh(ka*L)*sin(k*L));
        D1 = A1*(cosh(ka*L)*sin(k*L) + ka/k*sinh(ka*L)*cos(k*L));
        A = A1; C = C1; D = D1;
        psi(m1) = C1*cos(k*x(m1)) - D1*sin(k*x(m1));
        psi(m2) = A1*cosh(ka*x(m2));
        psi(m3) = C1*cos(k*x(m3)) + D1*sin(k*x(m3));
    else
        B2 = 1/sqrt(pi*(cosh(ka*L)^2+(ka/k)^2*sinh(k*L)^2));
        C2 = B2*(sinh(ka*L)*cos(k*L) - ka/k*cosh(ka*L)*sin(k*L));
        D2 = B2*(sinh(ka*L)*sin(k*L) + ka/k*cosh(ka*L)*cos(k*L));
        A = B2; C = C2; D = D2;
        psi(m1) =-C2*cos(k*x(m1)) + D2*sin(k*x(m1));
        psi(m2) = B2*sinh(ka*x(m2));
        psi(m3) = C2*cos(k*x(m3)) + D2*sin(k*x(m3));
    end
end
end

　　 画图

% === 设置参数 =====
m = 1; V0 = 1; E = 2; L = 2;
% =================
x = linspace(-5, 5, 1000);
psi_e = FSB(x, E, L, m, V0, 0);
psi_o = FSB(x, E, L, m, V0, 1);
figure; plot(x, real(psi_e)); hold on; plot(x, real(psi_o));
xlabel x; legend('\psi_{even}', '\psi_{odd}');

结果如图 1 。

1. 行波

% m 质量，V0 势垒，E 能量， L 势垒半宽， odd 奇函数 true 偶函数 false
% A,B 左区间系数， C,D 中区间系数， E,F 右区间
% pm = 1: 从左入射; pm = -1: 从右边入射
function [psi, A,B,C,D,E,F] = FSB2(x, E, L, m, V0, pm)
k = sqrt(2*m*E);
psi = zeros(size(x));
m1 = (x < -L); m2 = (abs(x) <= L); m3 = (x > L);
if (E == V0), E = V0 + eps; end
if E >= V0
    b = sqrt(2*m*(E-V0));
    N = ((k^2+b^2)*sin(2*b*L)-2i*k*b*cos(2*b*L))*exp(-1i*k*L)...
         /sqrt(2*pi)/((k^2-b^2)^2*sin(2*b*L)^2 + 4*k^2*b^2);
    A = 1/sqrt(2*pi);
    B = N*(k^2-b^2)*sin(2*b*L)*exp(-1i*k*L);
    C = N*1i*k*(b+k)*exp(-1i*b*L);
    D = N*1i*k*(b-k)*exp(1i*b*L);
    E = N*2i*k*b*exp(-1i*k*L);
    F = 0;
    if pm < 0
        tmp = A; A = F; F = tmp;
        tmp = B; B = E; E = tmp;
        tmp = C; C = D; D = tmp;
    end
    psi(m1) = A*exp(1i*k*x(m1)) + B*exp(-1i*k*x(m1));
    psi(m2) = C*exp(1i*b*x(m2)) + D*exp(-1i*b*x(m2));
    psi(m3) = E*exp(1i*k*x(m3)) + F*exp(-1i*k*x(m3));
else
    ka = sqrt(2*m*(V0-E));
    N = ((k^2-ka^2)*sinh(2*ka*L)-2i*k*ka*cosh(2*ka*L))*exp(-1i*k*L)...
         /sqrt(2*pi)/((k^2+ka^2)^2*sinh(2*ka*L)^2 + 4*k^2*ka^2);
    A = 1/sqrt(2*pi);
    B = N*(k^2+ka^2)*sinh(2*ka*L)*exp(-1i*k*L);
    C = N*k*(1i*ka-k)*exp(-ka*L);
    D = N*k*(1i*ka+k)*exp(ka*L);
    E = N*2i*k*ka*exp(-1i*k*L);
    F = 0;
    if pm < 0
        tmp = A; A = F; F = tmp;
        tmp = B; B = E; E = tmp;
        tmp = C; C = D; D = tmp;
    end
    psi(m1) = A*exp(1i*k*x(m1)) + B*exp(-1i*k*x(m1));
    psi(m2) = C*exp(ka*x(m2)) + D*exp(-ka*x(m2));
    psi(m3) = E*exp(1i*k*x(m3)) + F*exp(-1i*k*x(m3));
end
end

　　 画图

% === 设置参数 =====
m = 1; V0 = 0.9; E = 1.1; L = 2;
% =================
x = linspace(-5, 5, 1000);
psi = FSB2(x, E, L, m, V0, 1);
figure; plot(x, real(psi)); hold on; plot(x, imag(psi));
xlabel x;