Harmonic Oscillator¶
Quantarhei package (http://www.github.com/quantarhei)
ho (harmonic oscillator) module
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class
quantarhei.qm.oscillators.ho.fcstorage[source]¶ FC factor look-up class
Once Frank-Condon factors for some value of the shift are calculated they can be stored here, and retrieved when needed again
Methods
add(shift, fcmatrix)Adds the matrix of FC factors to the storage copy()Returns a shallow copy of the self deepcopy()Returns a deep copy of the self get(ii)Returns a stored FC matrix index(shift)Returns an index of the FC factors with a given shift load(filename[, test])Loads an object from a file and returns it loaddir(dirname)Returns a directory of objects saved into a directory lookup(shift)Returns true if the FC factors for a given shift are available save(filename[, comment, test])Saves the object with all its content into a file savedir(dirname[, tag, comment, test])Saves an object into directory containing a file with unique name scopy()Creates a copy of the object by saving and loading it
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class
quantarhei.qm.oscillators.ho.operator_factory(N=100)[source]¶ Class providing useful operators
Creation and anihilation operators
Methods
copy()Returns a shallow copy of the self deepcopy()Returns a deep copy of the self load(filename[, test])Loads an object from a file and returns it loaddir(dirname)Returns a directory of objects saved into a directory save(filename[, comment, test])Saves the object with all its content into a file savedir(dirname[, tag, comment, test])Saves an object into directory containing a file with unique name scopy()Creates a copy of the object by saving and loading it shift_operator(dd_)Calculates the Shift Operator based on the size N_ of the basis of states and the shift dd_. anihilation_operator creation_operator unity_operator -
shift_operator(dd_)[source]¶ Calculates the Shift Operator based on the size N_ of the basis of states and the shift dd_.
The shift operator is defined as
\[D_{\alpha} = e^{-\alpha \frac{\partial}{\partial Q}}\]where \(Q\) is the dimensionless coordinate of the Harmonic oscillator with Hamiltonian
\[H = \frac{\hbar\omega}{2}\left(P^2 + Q^2\right).\]In this definition, the shift operator acts on a statevector \(\psi(Q)\) in \(Q\)-representation is such a way that it shifts it by the value \(\alpha\) along the its coordinate \(Q\), i.e.
\[D_{\alpha}\psi(Q) = \psi(Q-\alpha).\]This definition is consistent with the definition of a shift operator as defined in Wikipedia (look for shift operator). It shifts the function to the right along the coordite \(Q\) axis (unlike in the definition in Wikipedia - this seems to be more natural for physicists.)
The dimensionless coordite \(Q\) and dimensionless momentum \(P\) are related to creation and annihilation operators as
\[a = \frac{1}{\sqrt{2}}\left(Q+iP\right)\]\[a^{\dagger} = \frac{1}{\sqrt{2}}\left(Q-iP\right)\]\[Q = \frac{1}{\sqrt{2}}\left(a + a^{\dagger}\right)\]\[P = \frac{1}{i\sqrt{2}}\left(a-a^{\dagger}\right).\]With these definitions we have
\[H = \hbar\omega\left(a^{\dagger}a + \frac{1}{2}\right)\]As we have \(P=-i\frac{\partial}{\partial Q}\), the shift operator reads as
\[D_{\alpha} = e^{-i\alpha P} = e^{-\frac{\alpha}{\sqrt{2}}\left(a-a^{\dagger}\right)}\]The operator can be generalized for complex values of \(\alpha\) to read
\[D_{\alpha} = e^{\frac{1}{\sqrt{2}}\left(\alpha a^{\dagger}-\alpha^{*}a\right)},\]where \(*\) represents complex conjugation. This is the definition implemented in Quantarhei.
The definition differs by the factor of \(\sqrt{2}\) from what is usually used in literature, but for applications in molecular physics, this definition seems to be more reasonable.
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class
quantarhei.qm.oscillators.ho.qrepresentation(qaxis)[source]¶ Coordinate representation of the HO wavefunctions
Methods
generate_ho_eigenfunctions()Generated q-representation of HO eigenfunctions get_coherent_state(alpha)Returns q-representation of a coherent state with a given alpha get_ho_eigenfunction(N)Returns q-representation of HO eigenfunction get_ho_ground_state()Returns the ground state wavefunction of the Harmonic oscillator get_probability_distribution(wfce)Returns probability distribution for a given wavefunction