Among the simplest applications of Schro¨dinger ’s time-independent equation for stationary states are the problems pertaining to the motion of a particle in the presence of potential steps. In the present chapter we shall consider the following problems: (a) Motion of a particle across a potential step of finite height and infinite extent.
Among the simplest applications of Schro¨dinger ’s time-independent equation for stationary states are the problems pertaining to the motion of a particle in the presence of potential steps. In the present chapter we shall consider the following problems: (a) Motion of a particle across a potential step of finite height and infinite extent.
Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
The solutions to the Time‐Independent Schrödinger Equation (TISE) can be found as long as we can define the Hamiltonian, which in the simplest cases, means defining the potential in which the particle of interest is moving. In the following sections, we will consider a single particle of mass, m p , moving in a potential that is time‐independent, but can vary in space. We will follow four steps: Define the Hamiltonian by defining the Potential Energy Operator since H ^ = - ħ 2 2 m p ∇ ^ 2 + V ^ ( r ) $ \hat{H} = - \frac{{{\hbar}^{2} }}{{2m_{p} }}\hat{\nabla }^{2} + \hat{V}(r) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315381381/5a219413-b541-4543-a0c3-fab4f5358e87/content/inline-math4_1.tif"/> . Solve the TISE to get the wave functions ψ ( x , y , z ) $ \psi (x, y, z) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315381381/5a219413-b541-4543-a0c3-fab4f5358e87/content/inline-math4_2.tif"/> and then the total stationary state wave functions Ψ ( x , y , z ; t ) = ψ ( x , y , z ) e - i ω t . $ \iPsi (x, y, z;t) = \psi (x, y, z)e^{ - i\omega t} . $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315381381/5a219413-b541-4543-a0c3-fab4f5358e87/content/inline-math4_3.tif"/> Determine the energies of the various states of the system. Examine the implications of the solutions to the Schrödinger equations.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
This chapter solves the eigen-problems for a few simple one-dimensional systems to acquire experience in solving the Schrodinger eigen-equation. One-dimensional problems are physically interesting not only because they are simple, but also because a number of more complicated higher-dimensional problems can be reduced to the solutions of equations in one dimension. For a symmetric potential, an eigenfunction of the Schrödinger equation can be always chosen to have a definite parity. The probability density of the plane wave solution is a constant, independent of the spatial coordinate of the particle. The eigensolution of a particle in an infinite square well consists of an infinite number of discrete energy levels. In classical mechanics, a particle cannot penetrate a potential barrier whose height is higher than the energy of the particle.