Oscillateur harmonique

Oscillateur harmonique

Nombre de niveaux       Visualise la phase       $$t$$ = $$\times 2\pi/\omega$$       $$\langle E \rangle$$ = $$\hbar\omega$$       $$\Delta x$$ = $$\sqrt{\hbar/(m\omega)}$$       $$\Delta p$$ = $$\sqrt{m\hbar\omega}$$       $$\Delta x \Delta p$$ = $$\hbar$$

This simulator shows the evolution of a 1D harmonic oscillator. The upper left panel shows the equidistant energy levels $$E_n = (n+1/2)\hbar\omega$$ and corresponding eigenstates $$\varphi_n(x)$$. The blue dot, centered on the average position $$\langle x\rangle$$, shows the corresponding classical motion. The upper right panel shows the coefficients $$c_n$$ of the decomposition of the initial state in the eigenbasis $$\{|\varphi_n\rangle\}$$. Using the mouse, you can adjust the amplitude of these coefficients, which are all supposed to be positive real quantities at $$t=0$$. Finally, the lower panel shows the probability distribution $$|\psi(x,t)|^2$$, or the complex wavefunction $$\psi(x,t)$$ depending on the state of the associated checkbox in the toolbar.
You can experiment different kinds of initial state : (i) a single eigenstate (or stationary state), (ii) a superposition of two specific eigenstates, (iii) a superposition of many different eigenstates. An interesting game consists of minimizing the product $$\Delta x \Delta p$$ by using a Poissonian distribution. If you get close enough to the lower bound of the Heisenberg inequality, you will observe that the corresponding wavefunction is gaussian. Furthermore, if you properly balance $$\Delta x$$ and $$\Delta p$$ (which corresponds to bringing the blue dot as close as possible to the average value of the energy), you can generate a quasi-classical state that will oscillate with minimal deformation. You can also more easily generate a (truncated) quasi-classical state simply by adjusting the average energy using the mouse in the upper left panel. Starting from such a state, you can then remove every other Hermite coefficient to get a Schrödinger cat.

How it's done

The initial state being decomposed in the eigenbasis, $|\psi(0)\rangle = \sum_n c_n |\varphi_n\rangle,$ the time evolution can be readily written as $|\psi(t)\rangle = \sum_n c_n e^{-i(n+1/2)\omega t}|\varphi_n\rangle.$ Using the well-known values of the Hermite functions, $$\varphi_n(x)$$, it is thus straightforward to compute the wavefuntion $\psi(x, t) = \sum_n c_n e^{-i(n+1/2)\omega t}\varphi_n(x)$ for each value $$t$$ of the time.
This simulator shows the evolution of a 1D harmonic oscillator. The upper left panel shows the equidistant energy levels $$E_n = (n+1/2)\hbar\omega$$ and corresponding eigenstates $$\varphi_n(x)$$. The blue dot, centered on the average position $$\langle x\rangle$$, shows the corresponding classical motion. The upper right panel shows the coefficients $$c_n$$ of the decomposition of the initial state in the eigenbasis $$\{|\varphi_n\rangle\}$$. Using the mouse, you can adjust the amplitude of these coefficients, which are all supposed to be positive real quantities at $$t=0$$. Finally, the lower panel shows the probability distribution $$|\psi(x,t)|^2$$, or the complex wavefunction $$\psi(x,t)$$ depending on the state of the associated checkbox in the toolbar.
You can experiment different kinds of initial state : (i) a single eigenstate (or stationary state), (ii) a superposition of two specific eigenstates, (iii) a superposition of many different eigenstates. An interesting game consists of minimizing the product $$\Delta x \Delta p$$ by using a Poissonian distribution. If you get close enough to the lower bound of the Heisenberg inequality, you will observe that the corresponding wavefunction is gaussian. Furthermore, if you properly balance $$\Delta x$$ and $$\Delta p$$ (which corresponds to bringing the blue dot as close as possible to the average value of the energy), you can generate a quasi-classical state that will oscillate with minimal deformation. You can also more easily generate a (truncated) quasi-classical state simply by adjusting the average energy using the mouse in the upper left panel. Starting from such a state, you can then remove every other Hermite coefficient to get a Schrödinger cat.

How it's done

The initial state being decomposed in the eigenbasis, $|\psi(0)\rangle = \sum_n c_n |\varphi_n\rangle,$ the time evolution can be readily written as $|\psi(t)\rangle = \sum_n c_n e^{-i(n+1/2)\omega t}|\varphi_n\rangle.$ Using the well-known values of the Hermite functions, $$\varphi_n(x)$$, it is thus straightforward to compute the wavefuntion $\psi(x, t) = \sum_n c_n e^{-i(n+1/2)\omega t}\varphi_n(x)$ for each value $$t$$ of the time.

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