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Harmonic oscillator

Number of levels Show 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.

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.

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.(c) 2019-2024 Manuel Joffre, all rights reserved