Final answer:
The expectation value of the square of the position operator in a quantum harmonic oscillator where the spring constant is set to zero takes a quantum mechanical approach, utilizing the time-dependent Schrödinger equation and considering the initial wave function and its time evolution.
Step-by-step explanation:
The problem in question involves the time dependence of the expectation value of the square of the position operator for a quantum harmonic oscillator with an initially disabled spring constant (spring constant set to zero). This scenario deviates from the simple harmonic motion typically discussed in classical mechanics, requiring a quantum approach for an accurate description.
In classical mechanics, when a system such as a block attached to a spring performs simple harmonic motion (SHM), the total energy at any time t is the sum of its potential energy and kinetic energy. For a classical oscillator with mass m and spring constant k, undergoing oscillations with amplitude A, the displacement x(t) as a function of time can be described using the equation:
x(t) = A cos[(√k/m) t].
However, when translating this into quantum mechanics, we need to compute the expectation value of the position squared, denoted <x²>, using the corresponding wave functions. Typically, due to symmetry, the expectation value of the position for a particle in the ground state of a harmonic oscillator is zero. This does not directly answer the student's question about the expectation value after the spring constant is disabled, but it provides context for applying quantum mechanical principles to the problem at hand.
In quantum mechanics, to find the time evolution of an observable after a sudden change (such as setting the spring constant to zero), one would generally use the time-dependent Schrödinger equation. Without additional details, we cannot provide a definitive answer to the question, but the method involves using the evolved wave function to compute the new expectation value <x²>(t) over time. Performing this calculation requires knowledge of the initial state of the system and its subsequent time evolution under the new potential, or lack thereof.