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In this section we will solve the Schrödinger equation fully for the first time, in order to learn about the quantum mechanics of a ‘‘free particle’’ — one where there are no forces acting, such that V (x) = 0 V (x)= 0. Before solving the Schrödinger equation, it is instructive to briefly recall how free particles behave in classical mechanics.
Practice problems – Schrodinger Equation and Atomic Physics 1. A particle is in the second excited state (n=3) in a one-dimentional square potential with absolutely impenetrable walls (0<x<L). Find the probability of the particle being in the region 1/3 L < x < 2/3 L. 2.
5.1 The Schr¨odinger and Heisenberg pictures Until now we described the dynamics of quantum mechanics by looking at the time evolution of the state vectors. This approach to quantum dynamics is called the Schrodinger picture.
where \( (H) \) and \( (S) \) stand for Heisenberg and Schrödinger pictures, respectively. The two operators are equal at \( t=0 \), by definition; \( \hat{A}^{(S)} = \hat{A}(0) \). On the other hand, in the Heisenberg picture the state vectors are frozen in time, \[ \begin{aligned} \ket{\alpha(t)}_H = \ket{\alpha(0)} \end{aligned} \]
We will focus mainly on the Schr ̈odinger equation to describe the evolution of a quantum-mechanical system. The statement that the evolution of a closed quantum system is unitary is however more general. It means that the state of a system at a later time t is given by |ψ(t)) = where U(t) is a unitary operator.
Schrodinger equation gives us a detailed account of the form of the wave functions or probability waves that control the motion of some smaller particles. The equation also describes how external factors influence these waves.
We’ve solved some simple quantum mechanics problems! The P-I-B model is a good approximation for some important cases, e.g. pi-bonding electrons on aromatics. Electronic transitions shift to lower energies as molecular size increases !
