Abstract
In this chapter we study simple quantum systems. A particle in a one-dimensional potential is described by a wave packet which is a solution of the time dependent Schrödinger equation. We discuss two approaches to discretize the second derivative. Finite differences are simple to use but their dispersion deviates largely from the exact relation, except high order differences are used. Pseudo-spectral methods evaluate the kinetic energy part in Fourier space and are much more accurate. The time evolution operator can be approximated by rational expressions like Cauchy’s form which corresponds to the Crank-Nicholson method. These schemes are unitary but involve time consuming matrix inversions. Multistep differencing schemes have comparable accuracy but are explicit methods. Best known is second order differencing. Split operator methods approximate the time evolution operator by a product. In combination with finite differences for the kinetic energy this leads to the method of real-space product formula which can be applied to wavefunctions with more than one component, for instance to study transitions between states. In a computer experiment we simulate a one-dimensional wave packet in a potential with one or two minima. Few-state systems are described with a small set of basis states. Especially the quantum mechanical two-level system is often used as a simple model for the transition between an initial and a final state due to an external perturbation. Its wavefunction has two components which satisfy two coupled ordinary differential equations for the amplitudes of the two states. In several computer experiments we study a two-state system in an oscillating field, a three-state system as a model for superexchange, the semiclassical approximation and the Landau-Zener model for curve-crossing and the ladder model for exponential decay. The density matrix formalism is used to describe a dissipative two-state system in analogy to the Bloch equations for nuclear magnetic resonance. In computer experiments we study the resonance line and the effects of saturation and power broadening. Finally we simulate the generation of a coherent superposition state or a spin flip by applying pulses of suitable duration. This is also discussed in connection with the manipulation of a Qubit represented by a single spin.
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Notes
- 1.
For instance collisions or the electromagnetic radiation field.
- 2.
If, for instance the wavefunction depends on the coordinates of N particles, the scalar product is defined by \(<\psi _{n}|\psi _{n'}>=\int d^{3}r_{1}\cdots d^{3}r_{N}\psi _{n}^{*}(r_{1}\cdots r_{N})\psi _{n'}(r_{1}\cdots r_{N})\).
- 3.
They are often called the “coherence” of the two states.
- 4.
The Pade approximation (Sect. 2.4.1) of order [1, 1].
- 5.
For simplicity only the case of even M is considered.
- 6.
This basis is usually incomplete.
- 7.
So called Rabi oscillations.
- 8.
For a diatomic molecule, e.g. the nuclear coordinate is simply the distance R of the two nuclei.
- 9.
The matrix of this system corresponds to the Liouville operator.
- 10.
We assume \(\varDelta \ge 0\), such that the equilibrium value of \(z=\rho _{11}-\rho _{22}\) is negative. Eventually, the two states have to be exchanged.
- 11.
For instance \(T_{2}=2T_{1}\) for pure radiative damping.
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Problems
Problems
Problem 23.1 Wave Packet Motion
In this computer experiment we solve the Schroedinger equation for a particle in the potential V(x) for an initially localized Gaussian wave packet \(\psi (t=0,x)\sim \exp (-a(x-x_{0})^{2})\). The potential is a box, a harmonic parabola or a fourth order double well. Initial width and position of the wave packet can be varied.
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Try to generate the stationary ground state wave function for the harmonic oscillator
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Observe the dispersion of the wave packet for different conditions and try to generate a moving wave packet with little dispersion.
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Try to observe tunneling in the double well potential
Problem 23.2 Two-state System
In this computer experiment a two-state system is simulated. Amplitude and frequency of an external field can be varied as well as the energy gap between the two states (see Fig. 23.9).
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Compare the time evolution at resonance and away from it
Problem 23.3 Three-state System
In this computer experiment a three-state system is simulated.
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Verify that the system behaves like an effective two-state system if the intermediate state is higher in energy than initial and final states (see Fig. 23.13).
Problem 23.4 Ladder Model
In this computer experiment the ladder model is simulated. The coupling strength and the spacing of the final states can be varied.
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Check the validity of the exponential decay approximation (see Fig. 23.15)
Problem 23.5 Semiclassical Approximation
In this computer experiment we study the crossing between two states along a nuclear coordinate. The time dependent Schrödinger equation for a wave packet approaching the crossing region is solved numerically and compared to the semiclassical approximation.
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Study the accuracy of the semiclassical approximation for different values of coupling and initial velocity
Problem 23.6 Landau–Zener Model
This computer experiment simulates the Landau Zener model . The coupling strength and the nuclear velocity can be varied (see Fig. 23.19).
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Try to find parameters for an efficient crossing of the states.
Problem 23.7 Resonance Line
In this computer experiment a two-state system with damping is simulated. The resonance curve is calculated from the steady state occupation probabilities (see Figs. 23.23, 23.24).
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Study the dependence of the line width on the intensity (power broadening).
Problem 23.8 Spin Flip
The damped two-state system is now subject to an external pulsed field (see Figs. 23.26, 23.27, 23.28).
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Try to produce a coherent superposition state (\(\pi /2\) pulse) or a spin flip (\(\pi \) pulse).
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Investigate the influence of decoherence.
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Scherer, P.O.J. (2017). Simple Quantum Systems. In: Computational Physics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-61088-7_23
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DOI: https://doi.org/10.1007/978-3-319-61088-7_23
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