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 We studied on the preceding page that "the probability density that a particle will be found is equal to the square of the absolute value of the wave function". An experimental evidence of the validity of the above interpretation was given by Gamow's explanation on the alpha decay of atomic nuclei. The alpha decay of atomic nuclei could be explained by a strange phenomenon called tunnel effect. This was an effect inconceivable in the classical theory. Let us learn it below.

 ["State" in Quantum Mechanics] Before getting onto the main point of the tunnel effect, let us explain the concept of "state" in quantum mechanics. It is a basic concept by which all the informations on the dynamical system under consideration, e.g. the energy and momentum distributions of the system, the probability density of the particles in the system, and so on, can be obtained. All of these informations about the system under consideration must be contained in the wave function. So that the wave function is sometimes called the state function.

 [Stationary State] Let us consider a one-dimensional space for simplicity. The wave function of a free particle is shown by Eq. (6) on the page "1-3: The Schroedinger Equation" as This is a state with a constant energy E . Thereby we can consider that a wave function whose time factor is is a state of a constant energy E . Let us assume the wave function to be As discussed above, this state is of a constant energy E . Putting this into the Schroedinger equation we have a differential equation independent of time t as which determines the spatial part of the wave function. The above equation (4) is named the "time-independent Schroedinger equation" or sometimes called the "Schroedinger equation" simply. In the state represented by the wave function of the form of Eq. (2), the probability density of the particle is written Because this probability is independent of time t , this state is called a stationary state, which is just what we have met in Bohr's old quantum theory. The ground state and the excited states of hydrogen atom are of this type of states.

 [Range of Motion of a Particle -- Classical Mechanics] Let us see the vibration of a spring or a harmonic oscillator as shown in Fig. (A) below. Suppose that a mass m is attached to the spring and neglect the mass of the spring itself. Suppose that an x axis is arranged in the direction of the motion of the oscillator. Let the position of the natural length of the spring be x = 0. When an energy E is given to the mass m , it oscillates within the range between the lower bound A and the upper bound B. The energies of this oscillator are graphed in the following Fig. (B). When the spring is extended by x from the equilibrium position (natural length), then the force exerted on the mass is - kx , where k is the spring constant. At this point, the potential energy of the oscillator is The total energy of the spring, E , is the sum of the kinetic energy and the potential energy, i.e., For a given energy E , the possible range of the oscillation is as follows: This is the range between A and B on the x axis in Fig. (B). In Fig. (B), the yellow-colored part denotes the potential wall. The particle motion is confined to the inner part of the potential (the outside of the wall). The particle with energy E is reflected by the potential wall, and repeats a back-and-forth motion between A and B on the x axis.

 [Energy Eigenvalues, Eigenstates] Let us treat the same oscillator motion as discussed above in quantum mechanics. The quantum mechanical state of motion is determined by the solutions of the Schroedinger equation (4). In the present case, it becomes the following differential equation The square of the absolute value of the wave function, , denotes the probability density where the particle will be detected. So that must be finite over all the region of x. In order that the wave function satisfies this condition, the corresponding energy E should be restricted to some specialized values. While the energy E can take a continuous value in the classical mechanics, it cannot be so in quantum mechanics. The allowed energy in quantum mechanics is not continuous but discrete (step-like). Such discrete energies are called the eigenvalues and the corresponding states are called the eigenstates. We should pay attention to the reason why such eigenstates with discrete energy eigenvalues occur. The reason is that the particle motion accompanies the wave function, i.e., just due to the particle-wave duality. The requirement of finiteness and continuity of the wave function brings about the discrete energy eigenvalues. This property is never obtained in the classical mechanics and it is a surprising result. Namely, the stationary states in Bohr's old quantum theory can be derived just from the double nature of electrons. The energy eigenvalues of a harmonic oscillator is given by The allowed n is 0 or positive integers. These results are shown in the form of energy levels in the following Fig. (C).

 The eigenstates of a harmonic oscillator The eigenvalues obtained by solving the Schroedinger equation (9) which are denoted by the horizontal lines (levels). The thick solid curves show the corresponding wave functions.
 In this figure, the ground state, n = 0, is denoted by the symbol (0), and the excited states, n = 1, 2, 3, ... by (1), (2), (3), ... , respectively. In the case of a harmonic oscillator, the energy levels (horizontal lines in Fig. (C)) are at equal intervals. This is a distinctive feature of a harmonic oscillator. The thick solid curves in Fig. (C) show the wave function corresponding to each energy eigenvalue. These results have been obtained by solving the Schroedinger equation (9). It can be solved analytically but it is somewhat tedious to explain how to solve it, so that we omit here. If you want to study the details, refer to some other textbooks on quantum mechanics.

 [Range of Motion of a Particle -- Quantum Mechanics] In the classical mechanics, the particle motion is confined to the inner part of the potential (the outside of the potential wall) as shown in Fig. (B). The particle cannot penetrate into the potential wall in the classical mechanics; i.e., the range of the particle motion in the classical mechanics is just from A to B in Fig. (B). However, the situation in quantum mechanics is different. In Fig. (C), the wave function corresponding to each eigenstate goes beyond the limits in the classical mechanics and it penetrates into the potential wall. This means that there is a probability that the particle will be found inside the potential wall. This is a surprising fact that we cannot imagine in the classical mechanics.

 [What's Tunnel Effect?] As discussed above, a particle can penetrate into a potential wall. We then suppose that a particle might pass through a potential wall (or barrier). That is true. This strange phenomenon is called "tunnel effect". The wave function of a free particle written as is a wave going to the positive direction on x axis with energy E (right-going wave). And the wave function is a left-going wave with the same energy. Let us consider a potential barrier as shown in Fig. (D) or Fig. (E). Suppose that an incident particle with energy E coming from the left collides the potential barrier. The energy E is assumed to be lower than the top of the barrier. In the classical mechanics, the particle is completely reflected by the potential barrier. In quantum mechanics, some probability will be reflected, but the other penetrates the barrier and pass through into the right-hand side region to proceed to the far right. This is the tunnel effect.

 [Penetration Coefficient] In order to estimate the rate of the probability that penetrates the potential barrier, let us assume a simple square potential as shown in the above Fig. (E). Here, let the width of the square potential be a and the height V0. The wave function in the region on the left-hand side of the barrier is a superposition of the incident wave (11) and the reflected wave (12); i.e., Contrarily, the wave function on the right-hand side of the barrier consists only of the the penetrated wave as Solving the Schroedinger equation, we can determine the constants A , B and C . Let us explain that on the following other page, 1-6-A: Calculation of the tunnel effect in a simple case. This page is not so difficult but, if you feel it tedious, you may pass it. Then you should accept the results. As seen in the calculation presented on the page 1-6-A, a relation holds. The quantity denotes the intensity of the incident wave. And and denote the intensities of the reflected and transmitted waves, respectively. The above relation (15) means therefore that the sum of the probabilities of the reflected particles and the transmitted particles is equal to 100 %. This is nothing but the conservation of the probability. The rate of the particles penetrating the potential barrier out of the incident particles, i.e. the penetration coefficient (or transmission coefficient) is given by where The function sinh defined by is called the hyperbolic sine function. An example of the experimental evidences of the tunnel effect discussed in the present section is the alpha decay of atomic nuclei which will be explained on the next page.
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