A particle of mass m in a one-dimensional harmonic oscillator is initially in a state given by |V(0)) = A (1) - |2)) where |n) are the time-independent eigenstates of the harmonic 1 oscillator with corresponding energy eigenvalues En = ħw(n+ } (a) Find the normalization constant A. (b) If one were to measure the energy of the particle in such state, what values would one get and what would be their associated probabilities? (c) Evaluate the time-dependence of the rms of the position operator:Ax = V(x²(t)) – (x)² |
A particle of mass m in a one-dimensional harmonic oscillator is initially in a state given by |V(0)) = A (1) - |2)) where |n) are the time-independent eigenstates of the harmonic 1 oscillator with corresponding energy eigenvalues En = ħw(n+ } (a) Find the normalization constant A. (b) If one were to measure the energy of the particle in such state, what values would one get and what would be their associated probabilities? (c) Evaluate the time-dependence of the rms of the position operator:Ax = V(x²(t)) – (x)² |
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![**Harmonic Oscillator**
A particle of mass \( m \) in a one-dimensional harmonic oscillator is initially in a state given by
\[ |\Psi(0)\rangle = A (|1\rangle - |2\rangle) \]
where \( |n\rangle \) are the time-independent eigenstates of the harmonic oscillator with corresponding energy eigenvalues \( E_n = \hbar \omega (n + \frac{1}{2}) \).
**(a)** Find the normalization constant \( A \).
**(b)** If one were to measure the energy of the particle in such a state, what values would one get and what would be their associated probabilities?
**(c)** Evaluate the time-dependence of the rms of the position operator:
\[ \Delta x \equiv \sqrt{\langle x^2(t) \rangle - \langle x \rangle^2} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2F4bd2a6e5-dad9-4229-9618-d6cc24d641d2%2Fxr11odg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Harmonic Oscillator**
A particle of mass \( m \) in a one-dimensional harmonic oscillator is initially in a state given by
\[ |\Psi(0)\rangle = A (|1\rangle - |2\rangle) \]
where \( |n\rangle \) are the time-independent eigenstates of the harmonic oscillator with corresponding energy eigenvalues \( E_n = \hbar \omega (n + \frac{1}{2}) \).
**(a)** Find the normalization constant \( A \).
**(b)** If one were to measure the energy of the particle in such a state, what values would one get and what would be their associated probabilities?
**(c)** Evaluate the time-dependence of the rms of the position operator:
\[ \Delta x \equiv \sqrt{\langle x^2(t) \rangle - \langle x \rangle^2} \]
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