Superposition state wavefunction

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

This spatial distribution is not stationary but evolves in time. So in this case, one has a wavefunction that is not a pure eigenstate of the Hamiltonian (one says that E is a superposition state or a non-stationary state) whose average energy remains constant (E=E2j lap + Ei,2 IbP) but whose spatial distribution changes with time. 

Since the wavefunction is not a stationary state, it evolves according to Equation 6.11. If there are only two stationary states in the superposition state, as in Figure 6.3, the probability distribution and all of the observables oscillate at a frequency a> = (If — E )/Ti (see Problem 6-10). If we have the left side wavefunction in Figure 6.3 at time / = 0, at later times we will have a right side wavefunction. At... 

Extension to many-particle systems is rather straight forward, and will be described later. It is physically obvious that j(q,t) should be zero (no current) for the stationary state wavefunction, which is a superposition... 

Due to the NAC, population is switched between the intersecting electronic states. Thereby, a superposition state and hence an electronic wavepacket is formed. In the vicinity of Coins, the time scales of the electron and nuclear dynamics are well synchronized. The energy difference between the coupled electronic states becomes very small slowing down the dynamics of the usually faster electrons to the time scale of the nuclear dynamics and below. The motion of the electronic density in the vicinity of a Coin is visualized in Fig. 8.14 for CoIn-1 of the cyclohexadiene/all-d -hexatriene system (Fig. 8.2). The underlying electronic wavepacket is created as the normalized superposition of the CASSCF-wavefunctions of ground and tirst excited state, keeping the nuclear geometry fixed. To describe the temporal evolution we take into account the time-dependent phase of both components. 

If and Tj are eigenfunctions of an operator, then any Unear combination C, Pj + CjTj also is an eigenfunction of that operator. A state represented by such a combined wavefunction is called a superposition state. One of the basic tenets of quantum mechanics is that we should use a linear combination of this nature to describe a system whenever we do not know which of its eigenstates the system is in. 

Although the probability of finding the particle at a given position varies with the position, the eigenfunctions given by Eq. (2.24) extend over the full length of the box. Wavefunctions for a particle that is more localized in space can be constructed from linear combinations of these eigenfunctions. A superposition state formed in... 

To see that the dipole moment of the superposition state oscillates with time, consider the same states at time t = (1 /2)h/ Eb — Ea). For the particle in a box, the energy of the second eigenstate is four times that of the first, Eb = AEa (Eq. 2.24), so ll2)hl Eb — Ea) = /6)h/Ea- The individual wavefunctions at this time have both real and imaginary parts. For the lower-energy state, we find by using the relationship exp(—/0) = cos (0) — i sin (0) that... 

Fig. 4.3 Wavefunction amplitudes (A, B) and probability densities (C, D) for two pure states and a superposition state. The dotted and dashed curves are for the first two eigenstates of an electron in a one-dimensional box of unit length, as given by Eq. (2.23) with n= or 2 and respectively). The solid curves are for the superposition 2 Ta + l b)- (A, C) The signed amplitudes of the wavefunctions and the probability densities at time t = 0, when all the wavefunctions are real. (B, D) The corresponding functions at time t = (ll2)h/ Ei, — Ea), where Ea and Ef, are the energies of the pure states...

It is an important result that even for time-independent Hamiltonian problems, stationary-state wavefunctions are not the only possible solutions of the TDSE. In fact, any arbitrary superposition (or linear combination) of different stationary-state wavefunctions is a solution of the TDSE. To illustrate this point, let Hq be some particular Hamiltonian with no explicit time dependence and let /, be the set of its spatial eigenfunctions with associated eigenvalues ,. ... 

Strictly speaking, a chiral species cannot correspond to a true stationary state of the time-dependent Schrodinger equation H

time scale for such spontaneous racemization is extremely long. The wavefunction of practical interest to the (finite-lived) laboratory chemist is the non-stationary Born-Oppenheimer model Eq. (1.2), rather than the true T of Eq. (1.1). 

When the fine structure frequencies fall below 100 MHz they can also be measured by quantum beat spectroscopy. The basic principle of quantum beat spectroscopy is straightforward. Using a polarized pulsed laser, a coherent superposition of the two fine structure states is excited in a time short compared to the inverse of the fine structure interval. After excitation, the wavefunctions of the two fine structure levels evolve at different rates due to their different energies. For example if the nd3/2 and nd5/2 mf = 3/2 states are coherently excited from the 3p3/2 state at time t = 0, the nd wavefunction at a later time t can be written as40... 

Taking the nuclear coordinates Q to be classical-like, the eikonal representation gives the wavefunction k(g, Q, t) for an initial electronic state / as a superposition of functions, of the form... 

Here k0 is the wave vector of the incident particle, x(K) is its spin wave function, and k is its spin coordinate. The second term in (4.3) is a superposition of products of scattered waves and the wavefunctions of all the possible molecular states that obey the energy conservation law ... 

In principle, with the scattering wavefunctions at one s disposal, it is possible to segment a complex superposition of single-ionization states in to portions belonging to different symmetries, channels, and spectral domains. Indeed,... 

Snider is best known for his paper reporting what is now referred to as the Waldmann-Snider equation.34 (L. Waldmann independently derived the same result via an alternative method.) The novelty of this equation is that it takes into account the consequences of the superposition of quantum wavefunctions. For example, while the usual Boltzmann equation describes the collisionally induced decay of the rotational state probability distribution of a spin system to equilibrium, the modifications allow the effects of magnetic field precession to be simultaneously taken into account. Snider has used this equation to explain a variety of effects including the Senftleben-Beenakker effect (i.e., is, the magnetic and electric field dependence of gas transport coefficients), gas phase NMR relaxation, and gas phase muon spin relaxation.35... 

The next step concerns the evaluation of the involved matrix elements. Within the lowest approximation, final-state channel interactions are neglected, and the many-electron wavefunctions are expressed as superpositions of Slater determin-antal wavefunctions with the correct symmetry and parity. In the present case of... 

Equation 6.8 does not always have to be satisfied f(x) does not have to be a stationary state. However, if f(x) does not satisfy Equation 6.8, the probability distribution P(x) and the expectation values of observables will change with time. The stationary states of a system constitute a complete basis set—which just means that any wavefunction f can be written as a superposition of the stationary states ... 

Consider two enantiomorphous molecules, R (right-handed) and L (left-handed), isolated from their surroundings and from external fields but not from each other. That is, they are allowed to interact. In the quantum-mechanical treatment of this system, as two particles in a one-dimensional symmetric double-well potential, the two states are degenerate in energy and are related to wavefunctions TR and 4Y localized in the two potential wells. Superposition of these wavefunctions yields ground and first excited states F+ and P ... 

In VB theory, all electrons are assigned to localized two-centered bonds, lone pairs and impaired components. Each collective pattern of such components constitutes a VB structure. The corresponding VB structure function can be expressed as < , namely, a bonded tableau (BT) state as mentioned before. The wavefunction for the whole system is defined as the superposition of all possible VB structure functions... 

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