\[\psi_{n}(x, t)=\sqrt{\frac{\alpha}{\sqrt{\pi} 2^{n} n !}} H_{n}(\alpha x) e^{-\frac{1}{2} \alpha^{2} x^{2}} e^{-i E_{n} t / \hbar} \quad(n=0,1,2,3, \ldots)\] The physical significance of the wave fu...\[\psi_{n}(x, t)=\sqrt{\frac{\alpha}{\sqrt{\pi} 2^{n} n !}} H_{n}(\alpha x) e^{-\frac{1}{2} \alpha^{2} x^{2}} e^{-i E_{n} t / \hbar} \quad(n=0,1,2,3, \ldots)\] The physical significance of the wave function is that its square, \(|\psi|^{2}=\psi^{*} \psi\), gives the probability of finding the particle at position \(x .{ }^{1}\) Squaring Eq. (60.8), we find this probability function for the harmonic oscillator is
The simple harmonic oscillator, a nonrelativistic particle in a quadratic potential , is an excellent model for a wide range of systems in nature. In fact, not long after Planck’s discovery that the b...The simple harmonic oscillator, a nonrelativistic particle in a quadratic potential , is an excellent model for a wide range of systems in nature. In fact, not long after Planck’s discovery that the black body radiation spectrum could be explained by assuming energy to be exchanged in quanta, Einstein applied the same principle to the simple harmonic oscillator, thereby solving a long-standing puzzle in solid state physics—the mysterious drop in specific heat of all solids at low temperatures.
