Harmonic Oscillator Done

Harmonic-Oscillator-Integrals.py

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+#!/usr/bin/env python3
+
+from sympy import *
+from sympy.physics.quantum.constants import hbar
+from sympy.physics.quantum.operator import DifferentialOperator
+from sympy.physics.quantum.qapply import qapply
+from sympy.physics.quantum.state import Wavefunction
+
+from sympy.physics.qho_1d import psi_n
+
+#Some general definitions
+m = Symbol('m', nonzero=True, positive=True)
+omega = Symbol('omega', nonzero=True, positive=True)
+x_var = Symbol('x')
+f = Function('f')
+
+########################################################################
+# This function calculates Bra Ket Products of form <n1| Operator |n2> #
+# for the harmonic 1d Oscillator with potential V(x) = 1/2 omega^2 x^2 #
+########################################################################
+
+def ho1dExpectaion(op, n1, n2):
+
+	def H(n,y):
+		yDiff = Symbol('yp')
+		out = (-1)**n
+		out *= exp(y**2)
+		out *= diff(exp(-(yDiff)**2), yDiff, n).subs(yDiff, y)
+		
+		return out
+
+	def h1dWaveFunction(n, y):
+		out = ((m*omega)/(pi *hbar))**Rational(1/4)
+		out *= 1/sqrt((2**n)*factorial(n))
+		out *= H(n,sqrt((m*omega)/hbar)*y)
+		out *= exp(-((m*omega)/(2*hbar))*y**2)
+
+		return out
+
+	Hbc = DifferentialOperator(Derivative(f(x_var),x_var), f(x_var))
+
+	w1 = Wavefunction(h1dWaveFunction(n1,x_var),x_var)
+	w2 = Wavefunction(h1dWaveFunction(n2,x_var),x_var)
+	v = qapply(op*w2)
+
+	return integrate(w1(x_var)*v(x_var), (x_var,-oo,oo))
+
+
+##################################
+# Here are some useful Operators #
+##################################
+
+#Momentum Operator
+p = DifferentialOperator(- I *hbar*Derivative(f(x_var),x_var), f(x_var))
+
+#Position Operator
+x = DifferentialOperator( x_var *f(x_var), f(x_var))
+
+#Hamiltonian
+H = DifferentialOperator(- hbar**2/(2*m) * Derivative(f(x_var),x_var,2) + (m/2) *omega**2 *x_var**2 *f(x_var), f(x_var))
+
+#Potential Energy Operator
+E_pot = DifferentialOperator((m/2) *omega**2 *x_var**2 *f(x_var), f(x_var))
+
+#Kinetic Energy Operator
+E_kin = DifferentialOperator(- hbar**2/(2*m) * Derivative(f(x_var),x_var,2), f(x_var))
+
+#Lowering Operator
+a = DifferentialOperator(sqrt((m * omega)/(2 * hbar))*(x_var*f(x_var) + ((hbar * Derivative(f(x_var),x_var))/(m * omega))), f(x_var))
+
+#Raising Operator
+ad = DifferentialOperator(sqrt((m * omega)/(2 * hbar))*(x_var*f(x_var) - (hbar * Derivative(f(x_var),x_var))/(m*omega)), f(x_var))
+
+
+
+#################################
+# Here you do your Calculations #
+#################################
+
+
+#Calculate for example <0| H |0>
+pprint(ho1dExpectaion(H,0,0))
+
+
+