+ iR^RIHtHt^RIHt^RIHt^RIHt^RI H t Rt Rt R#) )Ipi)S)exp)sqrt)hbarc\V4\V4r\V\,V,4\^\,4, #)aR Returns the wavefunction for particle on ring. Parameters ========== n : The quantum number. Here ``n`` can be positive as well as negative which can be used to describe the direction of motion of particle. x : The angle. Examples ======== >>> from sympy.physics.pring import wavefunction >>> from sympy import Symbol, integrate, pi >>> x=Symbol("x") >>> wavefunction(1, x) sqrt(2)*exp(I*x)/(2*sqrt(pi)) >>> wavefunction(2, x) sqrt(2)*exp(2*I*x)/(2*sqrt(pi)) >>> wavefunction(3, x) sqrt(2)*exp(3*I*x)/(2*sqrt(pi)) The normalization of the wavefunction is: >>> integrate(wavefunction(2, x)*wavefunction(-2, x), (x, 0, 2*pi)) 1 >>> integrate(wavefunction(4, x)*wavefunction(-4, x), (x, 0, 2*pi)) 1 References ========== .. [1] Atkins, Peter W.; Friedman, Ronald (2005). Molecular Quantum Mechanics (4th ed.). Pages 71-73. )rrrrr)nxs&&s/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/physics/pring.py wavefunctionr s3R Q41q q1uqy>DRL ((c\V4\V4\V4r!pVP'd8V^,\^,,^V,V^,,, #\R4h)ae Returns the energy of the state corresponding to quantum number ``n``. E=(n**2 * (hcross)**2) / (2 * m * r**2) Parameters ========== n : The quantum number. m : Mass of the particle. r : Radius of circle. Examples ======== >>> from sympy.physics.pring import energy >>> from sympy import Symbol >>> m=Symbol("m") >>> r=Symbol("r") >>> energy(1, m, r) hbar**2/(2*m*r**2) >>> energy(2, m, r) 2*hbar**2/(m*r**2) >>> energy(-2, 2.0, 3.0) 0.111111111111111*hbar**2 References ========== .. [1] Atkins, Peter W.; Friedman, Ronald (2005). Molecular Quantum Mechanics (4th ed.). Pages 71-73. z'n' must be integer)r is_integerr ValueError)r mrs&&&r energyr5sTJdAaD!A$!A|||1tQw1q51a4<00.//rN) sympy.core.numbersrrsympy.core.singletonr&sympy.functions.elementary.exponentialr(sympy.functions.elementary.miscellaneousrsympy.physics.quantum.constantsrr rrr rs&"690*)Z)0r