- #1
Haorong Wu
- 415
- 90
- Homework Statement
- Consider a particle subject to a one-dimensional simple harmonic oscillator potential. Suppose that at ##t=0## the state vector is given by
##exp \left ( \frac {-ipa} { \hbar } \right ) \left | 0 \right >##
where ##\left | 0 \right >## is one for which ##\left < x \right >= \left < p \right > =0##, ##p## is the momentum operator and ##a## is some number with dimension of length. Using the Heisenberg picture, evaluate the expectation value ##\left < x \right >## for ## t \ge 0 ##.
- Relevant Equations
- ##\frac {d A^{\left ( H \right )}} {dt} = \frac {1} {i \hbar} \left [ A^{\left ( H \right )} , H \right ]##
I have some problems when calculating the operators in Heisenberg picture.
First, ##\frac {dx} {dt} = \frac {1} {i \hbar} \left [ x, H \right ] = \frac {p} {m}##.
Similarly, ##\frac {dp} {dt} = \frac {1} {i \hbar} \left [ p, H \right ] = - m \omega ^ 2 x##.
These are coupled equations. I solved them and had
##x\left ( t \right ) = c_1 e^{-i \omega t} + c_2 e^{i \omega t}##, and ##p\left ( t \right ) = I am \omega \left ( - c_1 e^{-i \omega t} + c_2 e^{i \omega t} \right ) + c_3##.
Suppose the initial conditions are ##x\left ( 0 \right ) =x_0## and ## p \left ( 0 \right ) = p_0##.
Then I am still missing one extra condition to determine the three coefficients ##c_1##, ##c_2## and ##c_3##.
I am not sure what is the third condition. Maybe there just have to be one free coefficient?
Thanks!
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I am so sorry. I made a mistake. The coefficient ##c_3## should be deleted. How can I delete this post?
First, ##\frac {dx} {dt} = \frac {1} {i \hbar} \left [ x, H \right ] = \frac {p} {m}##.
Similarly, ##\frac {dp} {dt} = \frac {1} {i \hbar} \left [ p, H \right ] = - m \omega ^ 2 x##.
These are coupled equations. I solved them and had
##x\left ( t \right ) = c_1 e^{-i \omega t} + c_2 e^{i \omega t}##, and ##p\left ( t \right ) = I am \omega \left ( - c_1 e^{-i \omega t} + c_2 e^{i \omega t} \right ) + c_3##.
Suppose the initial conditions are ##x\left ( 0 \right ) =x_0## and ## p \left ( 0 \right ) = p_0##.
Then I am still missing one extra condition to determine the three coefficients ##c_1##, ##c_2## and ##c_3##.
I am not sure what is the third condition. Maybe there just have to be one free coefficient?
Thanks!
---------------------------------------------------
I am so sorry. I made a mistake. The coefficient ##c_3## should be deleted. How can I delete this post?
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