## Time-dependent Quantum Harmonic Oscillator

The system is described by a Hamiltonian that depends on time as shown in equation $$
\widehat{H} = \frac{\widehat{p}^{\,2}}{2} + \frac{\left(\widehat{q} - q_0(t)\right)^2}{2},$$ where $\widehat{p}$ and $\widehat{q}$ are the momentum and position operators respectively, and $q_0(t) = t/T$ where $t \in [0, T]$.

The ground state $\ket{\psi_0}$ of the time-independent harmonic oscillator is evolved in time using the split order method. For a time-independent Hamiltonian, the ground state $\ket{\psi(t)}$ can be obtained by applying the time evolution operator $e^{-i\widehat{H}t}$ to the initial state $\ket{\psi_0}$, such that: $$ \ket{\psi(t)} = e^{-i\widehat{H}t} \ket{\psi_0}. $$ I can use the fact that, if I take a small enough time-step $dt$ in order to solve the system: I discretize the system. I can evolve the state using $$ \ket{\psi(t+dt)} \approx e^{-i\widehat{H}\dd t} \ket{\psi(t)}$$ However, for a time-dependent Hamiltonian, the time evolution operator cannot be computed directly, and the split order method is used instead.

The split order method involves approximating the evolution operator $e^{-i\widehat{H} dt}$ using the Baker-Campbell-Hausdorff formula. This approximation can be written as $$ e^{-i\widehat{H} dt} \simeq e^{-i\frac{\widehat{V}}{2} dt} e^{-i\frac{\widehat{T}}{2} dt} e^{-i\frac{\widehat{V}}{2} dt},$$ where $\widehat{T}=\widehat{p}^{,2}/2$ is the kinetic energy operator and $\widehat{V}=(\widehat{q}-q_0(t))^2/2$ is the potential energy operator. The time evolution of the ground state can be computed using equation

To solve the Schrödinger equation for this Hamiltonian, one can discretize the $x$ space and write the Hamiltonian as a tridiagonal matrix. The discretization of space is performed by dividing the spatial interval $[-L, L]$ into $N$ equal parts of width $dx$, where $L$ is the size of the system and $N$ is the number of grid points.

The discretization of $x$ space implies that the position $x_i$ is given by $x_i = i*dx$ where $i = 0, 1, 2, ..., N$. In this discretization, eache component of the Hamiltonian can be approximated by the following tridiagonal matrix:

This code simulates the time evolution of a quantum harmonic oscillator. The simulation is performed by numerically solving the time-dependent Schrödinger equation using the split order method and discretization.

The code uses the

The

The

The

The code includes several functions for calculating the expected time to complete a given number of iterations, displaying a progress bar with an estimate of the time remaining, normalizing a vector, and plotting the evolution of the wave function over time and the average position of the particle.

The ground state $\ket{\psi_0}$ of the time-independent harmonic oscillator is evolved in time using the split order method. For a time-independent Hamiltonian, the ground state $\ket{\psi(t)}$ can be obtained by applying the time evolution operator $e^{-i\widehat{H}t}$ to the initial state $\ket{\psi_0}$, such that: $$ \ket{\psi(t)} = e^{-i\widehat{H}t} \ket{\psi_0}. $$ I can use the fact that, if I take a small enough time-step $dt$ in order to solve the system: I discretize the system. I can evolve the state using $$ \ket{\psi(t+dt)} \approx e^{-i\widehat{H}\dd t} \ket{\psi(t)}$$ However, for a time-dependent Hamiltonian, the time evolution operator cannot be computed directly, and the split order method is used instead.

The split order method involves approximating the evolution operator $e^{-i\widehat{H} dt}$ using the Baker-Campbell-Hausdorff formula. This approximation can be written as $$ e^{-i\widehat{H} dt} \simeq e^{-i\frac{\widehat{V}}{2} dt} e^{-i\frac{\widehat{T}}{2} dt} e^{-i\frac{\widehat{V}}{2} dt},$$ where $\widehat{T}=\widehat{p}^{,2}/2$ is the kinetic energy operator and $\widehat{V}=(\widehat{q}-q_0(t))^2/2$ is the potential energy operator. The time evolution of the ground state can be computed using equation

$$
\psi(x,t+ dt) \simeq e^{-i\frac{\widehat{V}}{2} dt} \mathcal{F}^{-1} \left[ e^{-i\frac{p^2}{2} dt} \mathcal{F}\left[ e^{-i\frac{\widehat{V}}{2} dt} \psi(x,t)\right] \right],
$$

which involves a Fourier transform to obtain an eigenstate in the momentum representation, followed by an anti-Fourier transform to return to the position representation. The time evolution is performed iteratively by taking small time steps $ dt$ up to a maximum time $T$. To solve the Schrödinger equation for this Hamiltonian, one can discretize the $x$ space and write the Hamiltonian as a tridiagonal matrix. The discretization of space is performed by dividing the spatial interval $[-L, L]$ into $N$ equal parts of width $dx$, where $L$ is the size of the system and $N$ is the number of grid points.

The discretization of $x$ space implies that the position $x_i$ is given by $x_i = i*dx$ where $i = 0, 1, 2, ..., N$. In this discretization, eache component of the Hamiltonian can be approximated by the following tridiagonal matrix:

$$
H_{ij} = \left[\left(-\frac{m}{2 dx^2}\right) + \frac{m}{2} w^2 x_i^2\right] \delta_{ij} - \frac{m}{2 dx^2} \delta_{i,j-1} - \frac{m}{2 dx^2} \delta_{i,j+1}
$$

where $\delta_{ij}$ is the Kronecker delta, which is equal to $1$ if $i = j$ and $0$ otherwise.This code simulates the time evolution of a quantum harmonic oscillator. The simulation is performed by numerically solving the time-dependent Schrödinger equation using the split order method and discretization.

The code uses the

**library to work with matrices and vectors, the***'NumPy'***library to compute the eigenvalues and eigenvectors of a tridiagonal matrix, and the***'SciPy'***library to create graphs.***'Matplotlib'*The

**function creates the tridiagonal Hamiltonian matrix and calculates its eigenvalues and eigenvectors using***'tridiagonal_and_eigenproblem'***'s***SciPy***function.***'eigh_tridiagonal'*The

**function calculates the moment space representation of the wave function obtained from the eigenvector of the Hamiltonian. The function normalizes the wave function and calculates the values of the k moments, which are needed for the momentum space representation.***'momentum_space'*The

**function generates a graph of the wave function in space and time. Specifically, the function plots the wave function in space, the wave function in moment space, and the probability density in space and time.***'plot'*The code includes several functions for calculating the expected time to complete a given number of iterations, displaying a progress bar with an estimate of the time remaining, normalizing a vector, and plotting the evolution of the wave function over time and the average position of the particle.

The system is described by a Hamiltonian that depends on time as shown in equation $$
\widehat{H} = \frac{\widehat{p}^{\,2}}{2} + \frac{\left(\widehat{q} - q_0(t)\right)^2}{2},$$ where $\widehat{p}$ and $\widehat{q}$ are the momentum and position operators respectively, and $q_0(t) = t/T$ where $t \in [0, T]$.

The ground state $\ket{\psi_0}$ of the time-independent harmonic oscillator is evolved in time using the split order method. For a time-independent Hamiltonian, the ground state $\ket{\psi(t)}$ can be obtained by applying the time evolution operator $e^{-i\widehat{H}t}$ to the initial state $\ket{\psi_0}$, such that: $$ \ket{\psi(t)} = e^{-i\widehat{H}t} \ket{\psi_0}. $$ I can use the fact that, if I take a small enough time-step $dt$ in order to solve the system: I discretize the system. I can evolve the state using $$ \ket{\psi(t+dt)} \approx e^{-i\widehat{H}\dd t} \ket{\psi(t)}$$ However, for a time-dependent Hamiltonian, the time evolution operator cannot be computed directly, and the split order method is used instead.

The split order method involves approximating the evolution operator $e^{-i\widehat{H} dt}$ using the Baker-Campbell-Hausdorff formula. This approximation can be written as $$ e^{-i\widehat{H} dt} \simeq e^{-i\frac{\widehat{V}}{2} dt} e^{-i\frac{\widehat{T}}{2} dt} e^{-i\frac{\widehat{V}}{2} dt},$$ where $\widehat{T}=\widehat{p}^{,2}/2$ is the kinetic energy operator and $\widehat{V}=(\widehat{q}-q_0(t))^2/2$ is the potential energy operator. The time evolution of the ground state can be computed using equation

To solve the Schrödinger equation for this Hamiltonian, one can discretize the $x$ space and write the Hamiltonian as a tridiagonal matrix. The discretization of space is performed by dividing the spatial interval $[-L, L]$ into $N$ equal parts of width $dx$, where $L$ is the size of the system and $N$ is the number of grid points.

The discretization of $x$ space implies that the position $x_i$ is given by $x_i = i*dx$ where $i = 0, 1, 2, ..., N$. In this discretization, eache component of the Hamiltonian can be approximated by the following tridiagonal matrix:

This code simulates the time evolution of a quantum harmonic oscillator. The simulation is performed by numerically solving the time-dependent Schrödinger equation using the split order method and discretization.

The code uses the

The

The

The

The code includes several functions for calculating the expected time to complete a given number of iterations, displaying a progress bar with an estimate of the time remaining, normalizing a vector, and plotting the evolution of the wave function over time and the average position of the particle.

The ground state $\ket{\psi_0}$ of the time-independent harmonic oscillator is evolved in time using the split order method. For a time-independent Hamiltonian, the ground state $\ket{\psi(t)}$ can be obtained by applying the time evolution operator $e^{-i\widehat{H}t}$ to the initial state $\ket{\psi_0}$, such that: $$ \ket{\psi(t)} = e^{-i\widehat{H}t} \ket{\psi_0}. $$ I can use the fact that, if I take a small enough time-step $dt$ in order to solve the system: I discretize the system. I can evolve the state using $$ \ket{\psi(t+dt)} \approx e^{-i\widehat{H}\dd t} \ket{\psi(t)}$$ However, for a time-dependent Hamiltonian, the time evolution operator cannot be computed directly, and the split order method is used instead.

The split order method involves approximating the evolution operator $e^{-i\widehat{H} dt}$ using the Baker-Campbell-Hausdorff formula. This approximation can be written as $$ e^{-i\widehat{H} dt} \simeq e^{-i\frac{\widehat{V}}{2} dt} e^{-i\frac{\widehat{T}}{2} dt} e^{-i\frac{\widehat{V}}{2} dt},$$ where $\widehat{T}=\widehat{p}^{,2}/2$ is the kinetic energy operator and $\widehat{V}=(\widehat{q}-q_0(t))^2/2$ is the potential energy operator. The time evolution of the ground state can be computed using equation

$$
\psi(x,t+ dt) \simeq e^{-i\frac{\widehat{V}}{2} dt} \mathcal{F}^{-1} \left[ e^{-i\frac{p^2}{2} dt} \mathcal{F}\left[ e^{-i\frac{\widehat{V}}{2} dt} \psi(x,t)\right] \right],
$$

which involves a Fourier transform to obtain an eigenstate in the momentum representation, followed by an anti-Fourier transform to return to the position representation. The time evolution is performed iteratively by taking small time steps $ dt$ up to a maximum time $T$. To solve the Schrödinger equation for this Hamiltonian, one can discretize the $x$ space and write the Hamiltonian as a tridiagonal matrix. The discretization of space is performed by dividing the spatial interval $[-L, L]$ into $N$ equal parts of width $dx$, where $L$ is the size of the system and $N$ is the number of grid points.

The discretization of $x$ space implies that the position $x_i$ is given by $x_i = i*dx$ where $i = 0, 1, 2, ..., N$. In this discretization, eache component of the Hamiltonian can be approximated by the following tridiagonal matrix:

$$
H_{ij} = \left[\left(-\frac{m}{2 dx^2}\right) + \frac{m}{2} w^2 x_i^2\right] \delta_{ij} - \frac{m}{2 dx^2} \delta_{i,j-1} - \frac{m}{2 dx^2} \delta_{i,j+1}
$$

where $\delta_{ij}$ is the Kronecker delta, which is equal to $1$ if $i = j$ and $0$ otherwise.This code simulates the time evolution of a quantum harmonic oscillator. The simulation is performed by numerically solving the time-dependent Schrödinger equation using the split order method and discretization.

The code uses the

**library to work with matrices and vectors, the***'NumPy'***library to compute the eigenvalues and eigenvectors of a tridiagonal matrix, and the***'SciPy'***library to create graphs.***'Matplotlib'*The

**function creates the tridiagonal Hamiltonian matrix and calculates its eigenvalues and eigenvectors using***'tridiagonal_and_eigenproblem'***'s***SciPy***function.***'eigh_tridiagonal'*The

**function calculates the moment space representation of the wave function obtained from the eigenvector of the Hamiltonian. The function normalizes the wave function and calculates the values of the k moments, which are needed for the momentum space representation.***'momentum_space'*The

**function generates a graph of the wave function in space and time. Specifically, the function plots the wave function in space, the wave function in moment space, and the probability density in space and time.***'plot'*The code includes several functions for calculating the expected time to complete a given number of iterations, displaying a progress bar with an estimate of the time remaining, normalizing a vector, and plotting the evolution of the wave function over time and the average position of the particle.