def kChange(start, stop, step, show=False):
k_range = np.arange(start, stop, step)
E_list = energy2(k_range, hbar, m, sig)
Es_list = []
for i in k_range:
Psi_x = gauss_init(x, i, x0=x0, d=sig)
sch = Schrodinger(x, Psi_x, V_x, hbar=hbar, m=m, t=0)
Es_list.append(sch.energy())
Diff_list = [Es_list[j] - E_list[j] for j in range(len(k_range))]
plt.figure()
plt.plot(k_range, E_list, label="Theoretical")
plt.plot(k_range, Es_list, label="Simulated", linestyle="--")
plt.title("k0")
plt.legend()
plt.xlabel("k0")
plt.ylabel("Energy")
plt.savefig("k0.png")
if show:
plt.show()
plt.figure()
plt.plot(k_range, Diff_list, label="Diff")
plt.title("k0_Diff")
plt.legend()
plt.xlabel("k0")
plt.ylabel("k0_Difference")
plt.savefig("k0_diff.png")
if show:
plt.show()
def run(Ns, step, dt, acc, c):
args = (m, omega, acc, x0, c)
v = harmonic_potential(x, t, args=args)
sch = Schrodinger(x, psi_x, v, k, hbar=hbar, m=m, t=0, args=args)
for i in range(Ns):
sch.evolve_t(N_steps=step, dt=dt)
return sch.expectation_x()
def T_g(V0, omeg):
v = gaussianBarrier(x, 0, omeg, V0)
E = scale * 10**3
k0 = np.sqrt(2 * m * E / (hbar**2) - 1 / (4 * sig**2))
psi_x = gauss_init(x, k0, x0, d=sig)
sch = Schrodinger(x, psi_x, v, hbar=hbar, m=m, t=0, args=L / 2)
return sch.impedence(E=E)
def varyE(E_list):
temp = []
for e in E_list:
print(e / 10 / bar_amp)
cb = [cosineBarrier(i, L, bar_amp) for i in x]
Sch = Schrodinger(x, x, cb, hbar=hbar, m=m)
temp.append(Sch.impedence(E=e))
return temp
def transmissionSpectrum(min, max, points, V):
sch = Schrodinger(x, Psi_x, V, hbar=hbar, m=m, t=0)
energies = np.linspace(min, max, points) * bar_amp
transmissions = []
for e in energies:
imp = sch.impedence(e)
transmissions.append(imp)
return energies, transmissions
def T_s(V0, E, gauss=False):
if gauss:
v = gaussianBarrier(x, 0, w, V0)
else:
v = squareBarrier(x, V0, -L / 2, L / 2)
k0 = np.sqrt(2 * m * E / (hbar**2) - 1 / (4 * sig**2))
psi_x = gauss_init(x, k0, x0, d=sig)
sch = Schrodinger(x, psi_x, v, hbar=hbar, m=m, t=0, args=L / 2)
return sch.impedence(E)
def T_t(l, V0, norm=False):
if norm:
v = tanhBarrierNorm(x, V0, l, w)
else:
v = tanhBarrier(x, V0, l, w)
psi_x = gauss_init(x, k0, x0, d=sig)
sch = Schrodinger(x, psi_x, v, hbar=hbar, m=m, t=0)
# plt.plot(x,v/scale)
# plt.show()
return sch.impedence(E)
def varyV(V_list, gauss=False):
temp = []
for v in V_list:
print(v / bar_amp)
if gauss:
cb = [gaussian(i, L, v) for i in x]
else:
cb = [cosineBarrier(i, L, v) for i in x]
Sch = Schrodinger(x, x, cb, hbar=hbar, m=m)
temp.append(Sch.impedence(E=E))
return temp
def runAll(self, N_step=1):
psi_comb = np.zeros((self.n, len(self.x)), dtype=complex)
for initial in self.initarray:
x0, v0, a0, g0 = initial
psi_x = self.hellerPacket(x0, v0, a0, g0)
sch = Schrodinger(self.x, psi_x, self.v, hbar=self.hbar, m=self.m)
for i in range(self.n):
if i != 0:
sch.evolve_t(N_step, self.dt)
psi_sol = sch.psi_x
psi_comb[i] += psi_sol / np.sqrt(self.N_packet)
self.psi_comb = psi_comb
return psi_comb
def run(acc):
t = 0
psi_x = gauss_init(x, k_initial, x0=0, d=d)
V_x = gaussbox(x, w, L, x0=xb, A=Amp, a=acc)
sch = Schrodinger(x, psi_x, V_x, k, hbar=hbar, m=m, t=t)
t_list = []
x_list = []
for i in range(Ns):
sch.evolve_t(N_steps=step, dt=dt)
t_list.append(sch.t)
x_list.append(sch.expectation_x())
dat = list(zip(t_list, x_list))
# np.savetxt("GBox_{}.txt".format(a), dat)
return dat
def Nchange(nrange, show=False):
E_ = energy2(k0, hbar, m, sig)
Es_list = []
for i in nrange:
N = 2**i
x = np.array([i * dx for i in range(N)])
x = x - max(x) / 2
Psi_x = gauss_init(x, k0, x0=x0, d=sig)
V_x = np.zeros(N)
sch = Schrodinger(x, Psi_x, V_x, hbar=hbar, m=m)
Es_list.append(sch.energy())
Diff_list = [Es_list[j] - E_ for j in range(len(nrange))]
plt.figure()
plt.plot(nrange, [E_ for i in nrange], label="Theoretical")
plt.plot(nrange, Es_list, label="Simulated", linestyle="--")
plt.title("N")
plt.legend()
plt.xlabel("N")
plt.ylabel("Energy")
plt.savefig("N.png")
if show:
plt.show()
plt.figure()
plt.ylim()
plt.plot(nrange, Diff_list, label="Diff")
plt.title("N_Diff")
plt.legend()
plt.xlabel("N")
plt.ylabel("Energy_Difference")
plt.savefig("N_diff.png")
if show:
plt.show()
Rel_list = [Diff_list[j] / E_ for j in range(len(nrange))]
plt.figure()
plt.plot(nrange, Rel_list, label="Relative Difference")
plt.title("N Relative Difference")
plt.legend()
plt.xlabel("N")
plt.ylabel("Relative_Energy_Difference")
plt.savefig("N_rel.png")
def mChange(start, stop, step, show=False):
m_range = np.arange(start, stop, step)
E_list = energy2(k0, hbar, m_range, sig)
Es_list = []
for i in m_range:
Psi_x = gauss_init(x, k0, x0=x0, d=sig)
sch = Schrodinger(x, Psi_x, V_x, hbar=hbar, m=i, t=0)
Es_list.append(sch.energy())
Diff_list = [Es_list[j] - E_list[j] for j in range(len(m_range))]
plt.figure()
plt.plot(m_range, E_list, label="Theoretical")
plt.plot(m_range, Es_list, label="Simulated", linestyle="--")
plt.title("M")
plt.legend()
plt.xlabel("M")
plt.ylabel("Energy")
plt.savefig("M.png")
if show:
plt.show()
plt.figure()
plt.plot(m_range, Diff_list, label="Diff")
plt.title("M_Diff")
plt.legend()
plt.xlabel("M")
plt.ylabel("Energy_Difference")
plt.savefig("M_diff.png")
if show:
plt.show()
Rel_list = [Diff_list[j] / E_list[j] for j in range(len(m_range))]
av = np.mean(Rel_list)
plt.figure()
plt.ylim(av - 0.1, av + 0.1)
plt.plot(m_range, Rel_list, label="Average.{}".format(round(av, 5)))
plt.title("M Relative Difference")
plt.legend()
plt.xlabel("M")
plt.ylabel("Relative_Energy_Difference")
plt.savefig("M_rel.png")
if show:
plt.show()
def sigChange(start, stop, step, show=False):
sig_range = np.arange(start, stop, step)
E_ = energy2(k0, hbar, m, sig_range)
Es_list = []
for i in sig_range:
Psi_x = gauss_init(x, k0, x0=x0, d=i)
sch = Schrodinger(x, Psi_x, V_x, hbar=hbar, m=m, t=0)
Es_list.append(sch.energy())
Diff_list = [Es_list[j] - E_[j] for j in range(len(sig_range))]
plt.figure()
plt.plot(sig_range, E_, label="Theoretical")
plt.plot(sig_range, Es_list, label="Simulated", linestyle="--")
plt.title("Sigma")
plt.legend()
plt.xlabel("Sigma")
plt.ylabel("Energy")
plt.savefig("Sigma.png")
if show:
plt.show()
plt.figure()
plt.plot(sig_range, Diff_list, label="Diff")
plt.title("Sigma_Diff")
plt.legend()
plt.xlabel("Sigma")
plt.ylabel("Energy_Difference")
plt.savefig("Sigma_diff.png")
if show:
plt.show()
Rel_list = [Diff_list[j] / E_[j] for j in range(len(sig_range))]
plt.figure()
plt.plot(sig_range, Rel_list, label="Relative Difference")
plt.title("Sigma Relative Difference")
plt.legend()
plt.xlabel("Sigma")
plt.ylabel("Relative_Energy_Difference")
plt.savefig("Sig_rel.png")
if show:
plt.show()
def run(x2, time=False):
V_x = barrier(x, A, x1, x2)
psi_init = gauss_init(x, k_init, x0=x0, d=sig)
sch = Schrodinger(x, psi_init, V_x, k, hbar=hbar, m=m, t=0, args=x2)
dat = []
t_list = []
for i in range(0, Ns):
sch.evolve_t(step, dt)
dat.append(sch.barrier_transmition())
t_list.append(sch.t)
# plt.plot(sch.k, abs(sch.psi_k))
# plt.show()
if time:
return dat, t_list
else:
return dat
def run(A, choen=False):
V_x = barrier(x, A, x1, x2)
sch = Schrodinger(x, Psi_x, V_x, hbar=hbar, m=m, args=x2, t=0)
I = sch.impedencePacket()
while sch.t < finalt:
print("Height :{h} Time:{t}".format(h=A, t=sch.t))
sch.evolve_t(step, dt)
T = sch.barrier_transmition()
return T, I
def run_A(A):
Psi_x = gauss_init(x, k0, x0=x0, d=sig)
V_x = gauss_barrier(x, A, x1, omeg)
sch = Schrodinger(x, Psi_x, V_x, hbar=hbar, m=m, t=0, args=x1)
imp = sch.impedencePacket(tol=10**-11)
time_list = []
Trans_list = []
while sch.t < t_final:
sch.evolve_t(N_steps=step, dt=dt)
time_list.append(sch.t)
Trans_list.append(sch.barrier_transmition())
print("Height {v}, Time {t}".format(v=A / scale, t=sch.t))
return sch.barrier_transmition(), imp
def onePacket(self, vkick, init):
nhalf = int(self.n / 2)
psi_arr = np.zeros((self.n, len(self.x)), dtype=complex)
x0, v0, a0, g0 = init
psi_init = self.hellerPacket(x0, v0, a0, g0)
tempsch = Schrodinger(self.x,
psi_init,
self.v,
hbar=self.hbar,
m=self.m)
for i in range(nhalf):
tempsch.evolve_t(1, dt=self.dt)
psi_arr[i] = tempsch.psi_x
tempsch.momentum_kick(self.m * vkick / self.hbar)
for i in range(nhalf):
tempsch.evolve_t(1, dt=self.dt)
psi_arr[i + nhalf] = tempsch.psi_x
return psi_arr
dt = 10**-7 dk = 2 * np.pi / (N * dx) k_lim = np.pi / dx k1 = -k_lim + (dk) * np.arange(N) # x = np.array([i * dx for i in range(N)]) x = np.arange(-lim, lim, dx) x0 = -1 * 10**-5 x1 = x[int(N / 2)] x2 = x1 + Len sig = 1 * 10**-6 k0 = 5 * 10**6 E = (hbar**2 / 2 * m) * (k0**2 + 1 / (4 * sig**2)) print(E) Psi_x = gauss_init(x, k0, x0=x0, d=sig) V_x = barrier(x, 10**-30, x1, x2) sch = Schrodinger(x, Psi_x, V_x, hbar=hbar, m=m, args=x1) # plt.plot(x, sch.psi_squared) plt.plot(x, V_x) plt.show() a = Animate(sch, V_x, 1000, dt, lim1=((-lim, lim), (0, max(sch.psi_squared)))) a.make_fig()
ks = fftshift(k)
# Defining time steps
t = 0
dt = 0.01
step = 1
Ns = 1000
print("Final time", dt * step * Ns)
hbar = 1
m = 1
args = (m, hbar)
p0 = k_initial * hbar
a0 = 1j * hbar / (4 * d**2)
g0 = (1j * hbar / 4) * np.log(2 * np.pi * d**2)
init = [x0, p0 / m, a0, g0]
psi_x = hellerPacket(x, x0, p0 / m, a0, g0)
sch = Schrodinger(x, psi_x, V_x, k, hbar=hbar, m=m)
hel = Heller(Ns, dt, init, derivs)
comp = SplitComp(sch, hel)
comp.packetComparisson(args)
# comp.comparrisonKick(args, kick=0)
# comp.comparrisonKick(args, kick=-2 * p0)
dx = 0.02
x_length = N * dx
x = np.linspace(0, x_length, N)
x1 = x[int(N / 4)]
x2 = x[int(3 * N / 4)]
# Defining Psi and V
k_initial = 4
psi_x = two_gauss(x, x1, x2, k_initial, -k_initial, d=1)
V_x = np.zeros(N)
# Defining K range
dk = dx / (2 * np.pi)
k = fftfreq(N, dk)
ks = fftshift(k)
# Defining time steps
t = 0
dt = 0.01
step = 2
sch = Schrodinger(x, psi_x, V_x, k)
# plt.plot(sch.x, sch.mod_square_x(True))
# plt.show()
a = Animate(sch, V_x, step, dt, lim1=((0, x_length), (0, max(np.real(psi_x)+0.2))),
lim2=((-2*k_initial, 2*k_initial), (0, abs(max(sch.psi_k)))), title='Two Packet Interferance')
a.make_fig()
# Barrier Definitions
A = 1
x1 = int(0.5 * N) * dx
W = 1
sig = 0.1
mu = 1
args = (A, W, x1, sig, mu)
k0 = 4
x0 = x[int(N / 4)]
sig = 4
k_init = 5
Psi_x = gauss_init(x, k0, x0=x0, d=sig)
sch = Schrodinger(x, Psi_x, noisyBarrier, hbar=hbar, m=m, args=args)
print("Width :", sig)
print(sch.x_width())
# print(sch.impedence())
dt = 0.001
step = 100
finalt = 40
c = Constants(A, dt, dx, k0)
a = Animate(sch, noisyBarrier, step, dt)
a.make_fig()
hbar = 1.0545718 * 10 ** -34
m = 1.44316072 * 10 ** -25
omegax = 80 * np.pi
sigma = 1 * 10 ** -6
t = 0
psi_init = wave_init(x, sigma, -x0)
V = harmonic_potential(x, m=m, w=omegax)
print(V[0])
plt.plot(x, psi_init)
plt.plot(x, V)
plt.show()
sch = Schrodinger(x, psi_init, V, hbar=hbar, m=m, t=0)
# a = Animate(sch, V, 1, dt,lim1=((-xmax, xmax), (0, max(sch.psi_squared))))
# a.make_fig()
simx = []
tl = []
El = []
temp = 0
while temp < Nt:
sch.evolve_t(1, dt=dt)
temp += 1
if (temp % nsave) == 0:
simx.append(sch.expectation_x())
# Defining K range
dk = dx / (2 * np.pi)
k = fftfreq(N, dk)
ks = fftshift(k)
# Defining time steps
t = 0
dt = 0.01
step = 5
Ns = 350
print(Ns * step * dt)
sch = Schrodinger(x,
psi_x,
harmonic_potential_t,
k,
hbar=hbar,
m=m,
t=t,
args=args)
# plt.plot(x, sch.mod_square_x(True))
# plt.plot(x, V_x)
# plt.ylim(0, max(np.real(psi_x)))
# plt.show()
#
# t_list = []
#
# expec_x = []
#
# for i in range(Ns):
# if i != 0:
sig = 4
k_init = 5
psi_init = gauss_init(x, k_init, x0=x0, d=sig)
#
dk = dx / (2 * np.pi)
k = fftfreq(N, dk)
ks = fftshift(k)
t = 0
dt = 0.01
step = 5
Ns = 600
print(dt * Ns * step)
sch = Schrodinger(x, psi_init, V_x, k, hbar=hbar, m=m, t=t)
################### Testing
# t_list = []
# for i in range(0, Ns):
# t += dt * step
# t_list.append(t)
# t = 0
#
# L_list = np.linspace(1, 6, 10)
# Tunnel_Val = []
#
# figure = plt.figure()
# ax = figure.add_subplot(1, 1, 1)
# ax.set_title("Tunneled Packet vs Time")
# ax.set_xlabel("Time")
((K**3 / (k1 * k2) + k1 * k2 / K) * np.sin(2 * K * a) - 1j * K *
(k1 / k2 + k2 / k1) * np.cos(2 * K * a)) * np.sinh(k2 * delta) *
np.sinh(k1 * delta))
if A == 0:
A = 10**-99
return 1 - abs(B / A)**2
Psi_x = gauss_init(x, k0, x0=x0, d=sig)
V_x = doubleBarrier(x, bar_amp, x1, wid, sep)
# V_x = np.zeros(len(x))
sch = Schrodinger(x, Psi_x, V_x, hbar=hbar, m=m, t=0)
######## Initial Plot
# plt.plot(x / 10 ** -6, sch.psi_squared)
# plt.plot(x / 10 ** -6, V_x / scale)
# plt.show()
######## Animate
# a = Animate(sch, V_x, step, dt, lim1=((x[0], x[-1]), (0, max(np.real(sch.psi_squared)))),
# lim2=((sch.k[0], sch.k[-1]), (0, max(np.real(sch.psi_k)))))
# a.make_fig()
#############Ashby Tunneling Test
a = sep / 2
b = sep / 2 + wid
min = 0
return 1 - coeff
e = bar_amp
l = np.sqrt(hbar ** 2 / (2 * m * bar_amp))
L = 10 * l
x = np.linspace(-20, 20, 1000) * l
dx = x[1] - x[0]
v_sq = squareBarrier(x, bar_amp, -L / 2, L / 2)
v_tanh = tanhBarrier(x, bar_amp, L, 10 ** -8)
psi_x = gauss_init(x, k0, x0, d=sig)
sch = Schrodinger(x, psi_x, v_sq, hbar=hbar, m=m, t=0, args=L / 2)
ys = tanhBarrierNoNorm(x, bar_amp, L, 10 ** -6)
# ppot, pcov = curve_fit(gaussian, x, ys, p0=(0, 2*10 ** -6, bar_amp))
# plt.plot(x / 10 ** -6, ys / scale * 10 ** -3, label="Tanh")
# plt.plot(x / 10 ** -6, gaussian(x, ppot[0], ppot[1], ppot[2]) / scale * 10 ** -3, label="Fit")
# plt.legend()
# plt.xlim(-10, 10)
# plt.ylabel("V (kHz)")
# plt.xlabel("x (micrometers)")
# plt.title("Tanh and Gaussian Comparison")
# plt.savefig("Gaussian Comparison No Norm")
# plt.show()
# w_list = np.logspace(-6, -8, 5)
imp = sch.impedencePacket(tol=10**-11)
time_list = []
Trans_list = []
while sch.t < t_final:
sch.evolve_t(N_steps=step, dt=dt)
time_list.append(sch.t)
Trans_list.append(sch.barrier_transmition())
print("Height {v}, Time {t}".format(v=A / scale, t=sch.t))
return sch.barrier_transmition(), imp
Psi_x = gauss_init(x, k0, x0=x0, d=sig)
V_x = gauss_barrier(x, bar_amp, x1, omeg)
sch = Schrodinger(x, Psi_x, V_x, hbar=hbar, m=m, t=0, args=x1)
# print("Energy", sch.energy())
# print("Imp",sch.impedencePacket(tol=10**-11))
# plt.plot(sch.k, sch.mod_square_k(r=True))
# plt.plot(x, np.asarray(V_x)/scale)
# plt.show()
# a = Animate(sch, V_x, step, dt, lim1=((x[0], x[-1]), (0, max(np.real(sch.psi_squared)))),
# lim2=((sch.k[0], sch.k[-1]), (0, max(np.real(sch.psi_k)))))
# a.make_fig()
# v_list = np.arange(0.5, 5, 0.1)
################ Changing V
# v_list = np.linspace(0.1, 3, 50) * bar_amp
def tunnelingSpectra(V, E_list):
sch = Schrodinger(x, psi_x, V, hbar=hbar, m=m, t=0, args=L / 2)
T_list = []
for e in E_list:
T_list.append(sch.impedence(e))
return T_list
# Defining K range
dk = dx / (2 * np.pi)
k = fftfreq(N, dk)
ks = fftshift(k)
# Defining time steps
t = 0
dt = 0.01
step = 10
Ns = 100
print("Final time", dt * step * Ns)
hbar = 1
m = 1
sch = Schrodinger(x, psi_x, V_x, k, hbar=hbar, m=m)
print("Numerical Energy :", sch.energy())
print("Theoretical Energy", (hbar**2 * k_initial**2) / (2 * m))
print("Difference :", sch.energy() - (hbar**2 * k_initial**2) / (2 * m))
psi_init2 = gauss_init(x, k_initial, x0=x0, d=d)
psis2 = np.real(psi_init2 * np.conj(psi_init2))
# plt.plot(x, sch.mod_square_x(True))
# plt.plot(x, V_x)
# plt.ylim(0, max(np.real(psi_x)))
# plt.show()
a = Animate(sch,
def T_s(l, V0):
v = squareBarrier(x, V0, -l / 2, l / 2)
psi_x = gauss_init(x, k0, x0, d=sig)
sch = Schrodinger(x, psi_x, v, hbar=hbar, m=m, t=0)
return sch.impedence(E)
