Method for First Excited State - Edubirdie

University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

57

Pages:

5
Author:

Jabin F.

Method for first excited statte Choice of trial function:- Here change in no. of nodes compare to ground Stater. if 4. ground state 4,→ first excited state = 0 (i)) = = <Ψ|H|Ψ> Eexact it Trial = Yexact⇒ = Eexact. If 1 Trial ≠ Yexact > Eexact > Eexact If (in (2x) = Aen², = 2m² (17) Ψ(n) = AEFM, Prob. Calculate the ground state energy of a particle of mass 'm' moving in 1-D plot? V(x) = mw²/2x² by using variational method. Sol Ψ = - - Ae Ψ = Ae ^|-x| Ψ = Ae^(-αx²) Ψ = Ae^(-αx²) = = ∫Ψ* H Ψ dx ∫Ψ* Ψ dx H = (-ħ²/2m) d²/dx² + (mw²/2)x² = + = ∫₀∞ ħ²/2m (dψ/dx)² dx ∫₀∞ ψ² dx ħ²/2m ∫₀∞ (α²)² e^(-2αx²) dx ∫₀∞ e^(-2αx²) dx ħ²/2m α² ∫₀∞ e^(-y) dy ∫₀∞ e^(-y) dy ħ²α²/2m The equations involve various symbols, including numbers, Greek letters (alpha), and potentially other mathematical operators. The content is dense and complex, suggesting advanced mathematical or physics calculations. The writing is in black ink on white paper. There are no other elements or objects in the image. There are no visible drawings or diagrams. The writing is relatively neat, though some characters might be difficult to interpret accurately due to handwriting variations. The equations are aligned vertically, suggesting a structured problem-solving approach. ⇒ [\alpha^{2}=\frac{m^{2}\omega^{2}}{4~\hbar^{2}}] if (\alpha=\frac{-mu0}{\alpha^{\prime}\hbar}) 20 (\alpha=\frac{m\omega}{2~\hbar}) ": (\psi(\pi)=Ae^{-d\pi e^{2}}) ر [\psi\rightarrow\infty] [x\ne\infty] ( 20 (\langle E\rangle=\frac{\hbar^{2}}{2m}(\frac{m\omega}{2\hbar})+\frac{m\omega^{2}}{8(\frac{m\omega}{2\hbar})}) JZLES SO 222 [\alpha=\pm\frac{m~\omega}{2~\hbar}]

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