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6 Quantum particle in complex potentials In this section we consider how spatial confinements modify the motion and the wave function of a quantum particle. The reader is referred to Refs. [15–18] for more detail. A rectangular well with infinite walls The steady-state Shr¨odinger equation for a particle in a rectangular potential well with infinite walls (Fig. e. − U (x) = 0 ∞ for |x| < a/2 . 70) Based on the symmetry of the problem one can foresee odd and even solutions. The symmetry of the potential, U (x) = U (−x) results in a symmetry of the probability density, |ψ(x)|2 = |ψ(−x)|2 , whence, ψ(x) = ±ψ(−x), and so we arrive at two independent solutions with different parity.

Later, in 1932, he considered in detail the density of states in terms of the number of elementary phase space cells [12]. We now show how Rayleigh’s counting modes evolves to the Heisenberg uncertainty relation. e. x = L. e. k = π/L = k = π/ x. We arrive at the relation, x k = π. 49) Now using p = h¯ k (Eq. 51) with an accuracy of factor 2. To summarise, we have seen in this section that Rayleigh’s approach of counting modes in a finite cavity combined with the idea of wave properties of material particles, p = h¯ k, leads to the Heisenberg uncertainty relation and to Plank’s idea of a discrete cellular structure of the phase space on a microscale.

74) grows monotonically with n. For every state the wave function equals zero at the walls. The total probability of finding a particle inside the well equals 1. Note, Eq. 73) gives the values of kinetic energy. Using Eqs. 7) we can write expressions for particle momentum, wave number and wavelength: pn = π h¯ n, a kn = π n, a λn = 2a . 75) Note that wavelengths correspond to integer numbers of λ/2 inside the well. 76) 24 Basic properties of electromagnetic waves and quantum particles is referred to as the particle zero energy.