Quantum Mechanics Exam: Quiz!

In quantum mechanics, a particle in a box refers to a hypothetical scenario where a particle is confined within a finite potential well. According to the Heisenberg uncertainty principle, it is impossible for a particle to have both a well-defined position and momentum simultaneously. Therefore, a particle in a box can never be at rest, as this would require a precise measurement of both position and momentum, which is not allowed by quantum mechanics. Hence, the statement "Particle in a box of finite potential can never be at rest" is true.

The wave function of a particle in a box can be represented by a linear combination of sine and cosine functions, as given by Asin(kx) + Bcos(kx). This form allows for the representation of both the amplitude and phase of the wave function. The coefficients A and B determine the relative contributions of the sine and cosine components, allowing for a more general representation of the wave function.

The solution of the Schrodinger wave equation for the Tunnel effect is of the form Aeikx+ Be-ikx. This solution represents a wave traveling in the positive x-direction (Aeikx) and a wave traveling in the negative x-direction (Be-ikx). The presence of both positive and negative exponential terms indicates the possibility of wave transmission through a potential barrier, which is a characteristic of the Tunnel effect.

The zero-point energy for a particle in an infinite potential well is given by the equation E = (π^2 * h^2) / (2mL^2), where h is the Planck's constant, m is the mass of the particle, and L is the length of the well. In this case, we are given that the electron is confined to a 1 nm atom, so L = 1 nm. Plugging in the values into the equation, we get E = (π^2 * h^2) / (2 * m * (1 nm)^2). Since the values of h and m are constant, the only variable is L. Therefore, the zero-point energy will be directly proportional to L^2. As L^2 increases, the zero-point energy will also increase. Among the given options, the only value that is larger than the others is 5.9 X 10-29 J.

The transmission based on tunnel effect is that of a plane wave through a rectangular barrier.

In a finite potential well, the potential energy outside the box is constant. This means that the potential energy remains the same at all points outside the box, regardless of the distance from the well. This is because the potential energy in a finite potential well only changes within the region of the well itself, while outside the well, it remains constant.

In a simple harmonic motion, the magnitude of average acceleration in half a time period is given by 2ω²A/π. This can be derived from the equation of motion for simple harmonic motion, where acceleration is proportional to displacement and opposite in direction. The average acceleration is calculated by taking the difference in velocities at the beginning and end of half a time period and dividing it by the time taken. In this case, the average acceleration is found to be 2ω²A/π.

The correct answer is Heisenberg's uncertainty principle. The uncertainty principle states that it is impossible to simultaneously determine the exact position and momentum of a particle with absolute certainty. This principle is fundamental to the understanding of the tunnel effect, which is the phenomenon where particles can pass through barriers that would be classically impossible to penetrate. The uncertainty in the particle's position allows it to "tunnel" through the barrier, appearing on the other side even though it does not have enough energy to overcome the barrier classically.