Final answer:
To maximize (x) for a linear combination of |0) and (1) in a one-dimensional simple harmonic oscillator, we choose to align with the first excited state and set the coefficients such that c_0=0 and c_1=1.
Step-by-step explanation:
The student's question pertains to maximizing the expectation value of the position (x) for a combination of the ground state |0) and the first excited state (1) of a one-dimensional harmonic oscillator without using wave functions. The linear combination can be represented as |\psi) = c_0|0) + c_1|1), where c_0 and c_1 are complex coefficients that need to be determined.
Given that the expected value of (x) for the ground state is zero due to symmetry and the odd nature of the wave functions involved, we can maximize (x) by choosing a state that aligns with the first excited state that has a non-zero expectation value of (x). Therefore, by setting c_0 = 0 and c_1 = 1, we achieve the state with the maximum expectation value for position.
To construct a linear combination of |0) and (1) such that (x) is as large as possible, we need to find the maximum value of (x). Using the equation for the magnitude of velocity as a function of position for a simple harmonic oscillator, which is given as 101 = √√ = 14 (1²-x²), we can see that the velocity is zero at the two extreme positions, x = ±1.
Therefore, to maximize (x), we need to choose the linear combination of |0) and (1) such that the position is at one of the extreme points. In this case, (x) is as large as possible when we choose the linear combination of |0) and (1) with x = ±1.