Therefore, the probability density of finding the classical particle at x is uniform throughout the box, and there is no preferable location for finding a classical particle. Assuming that its speed u is constant, this time is Δ t = Δ x / u, Δ t = Δ x / u, which is also constant for any location between the walls. The probability density of finding a classical particle between x and x + Δ x x + Δ x depends on how much time Δ t Δ t the particle spends in this region. Stationary states are states of definite energy, but linear combinations of these states, such as ψ ( x ) = a ψ 1 + b ψ 2 ψ ( x ) = a ψ 1 + b ψ 2 (also solutions to Schrӧdinger’s equation) are states of mixed energy.įigure 7.12 The probability density distribution | ψ n ( x ) | 2 | ψ n ( x ) | 2 for a quantum particle in a box for: (a) the ground state, n = 1 n = 1 (b) the first excited state, n = 2 n = 2 and, (c) the nineteenth excited state, n = 20 n = 20. These functions are “stationary,” because their probability density functions, | Ψ ( x, t ) | 2 | Ψ ( x, t ) | 2, do not vary in time, and “standing waves” because their real and imaginary parts oscillate up and down like a standing wave-like a rope waving between two children on a playground. The wave functions in Equation 7.45 are also called stationary state s and standing wave state s. Energy levels are analogous to rungs of a ladder that the particle can “climb” as it gains or loses energy. The wave functions in Equation 7.45 are sometimes referred to as the “states of definite energy.” Particles in these states are said to occupy energy levels, which are represented by the horizontal lines in Figure 7.11. The first three quantum states (for n = 1, 2, and 3 ) n = 1, 2, and 3 ) of a particle in a box are shown in Figure 7.11. The state for n = 2 n = 2 is the first excited state, the state for n = 3 n = 3 is the second excited state, and so on. The index n is called the energy quantum number or principal quantum number. This physical situation is called the infinite square well, described by the potential energy function Between the walls, the particle moves freely. The energy of the particle is quantized as a consequence of a standing wave condition inside the box.Ĭonsider a particle of mass m m that is allowed to move only along the x-direction and its motion is confined to the region between hard and rigid walls located at x = 0 x = 0 and at x = L x = L ( Figure 7.10). This special case provides lessons for understanding quantum mechanics in more complex systems. In this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. Explain the physical meaning of Bohr’s correspondence principle.Describe the physical meaning of stationary solutions to Schrӧdinger’s equation and the connection of these solutions with time-dependent quantum states.Explain why the energy of a quantum particle in a box is quantized.Describe how to set up a boundary-value problem for the stationary Schrӧdinger equation.By the end of this section, you will be able to:
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