Return to Question

The following is a section "Correlation Amplitude and the Energy-Time Uncertainty Relation" from Sakurai's Modern Quantum Mechanics book page 79:

Question:

• Why does it state that the oscillations are rapid unless $|E-E_0|$ is small compared to $\frac{\hbar}{t}$? Also, what is the particular importance of the interval $|E-E_0| \approx \frac{\hbar}{t}$ being narrow compared to $\Delta E$ (defined as the width of $|g(E)|^2\rho(E)$) which I think allows us to assume that $|g(E)|^2 \rho(E)$ is approximately constant (which then implies that $C(t)$ is zero by allowing us to take $|g(E)|^2 \rho(E)$ out of the integral (2.1.72) and hence an integral on an oscillating function of that kind is zero)?

Thanks for any assistance.

The following is a section "Correlation Amplitude and the Energy-Time Uncertainty Relation" from Sakurai's Modern Quantum Mechanics book page 79:

Question:

• Why does it state that the oscillations are rapid unless $|E-E_0|$ is small compared to $\frac{\hbar}{t}$? Also, what is the particular importance of the interval $|E-E_0| \approx \frac{\hbar}{t}$ being narrow compared to $\Delta E$ (defined as the width of $|g(E)|^2\rho(E)$) which I think allows us to assume that $|g(E)|^2 \rho(E)$ is approximately constant (which then implies that $C(t)$ is zero by allowing us to take $|g(E)|^2 \rho(E)$ out of the integral (2.1.72) and hence an integral on an oscillating function of that kind is zero)?

The following is a section "Correlation Amplitude and the Energy-Time Uncertainty Relation" from Sakurai's Modern Quantum Mechanics book page 79:

Question:

• Why does it state that the oscillations are rapid unless $|E-E_0|$ is small compared to $\frac{\hbar}{t}$? What is the particular importance of the interval $|E-E_0| \approx \frac{\hbar}{t}$ being narrow compared to $\Delta E$ (width of $|g(E)|^2\rho(E)$) which allows us to assume that $|g(E)|^2 \rho(E)$ is constant (which then implies that $C(t)$ is zero)?

Thanks for any assistance.

The following is a section "Correlation Amplitude and the Energy-Time Uncertainty Relation" from Sakurai's Modern Quantum Mechanics book page 79:

Question:

• Why does it state that the oscillations are rapid unless $|E-E_0|$ is small compared to $\frac{\hbar}{t}$? Also, what is the particular importance of the interval $|E-E_0| \approx \frac{\hbar}{t}$ being narrow compared to $\Delta E$ (defined as the width of $|g(E)|^2\rho(E)$) which I think allows us to assume that $|g(E)|^2 \rho(E)$ is approximately constant (which then implies that $C(t)$ is zero by allowing us to take $|g(E)|^2 \rho(E)$ out of the integral (2.1.72) and hence an integral on an oscillating function of that kind is zero)?

Thanks for any assistance.

The following is a section "Correlation Amplitude and the Energy-Time Uncertainty Relation" from Sakurai's Modern Quantum Mechanics book page 79:

Question:

• How does it follow that if we take an interval $|E-E_0| \approx \frac{\hbar}{t}$ which is much narrower than $\Delta E$ (width of $|g(E)|^2\rho(E)$) then there is essentially no contribution to $C(t)$?

The following is a section "Correlation Amplitude and the Energy-Time Uncertainty Relation" from Sakurai's Modern Quantum Mechanics book page 79:

Question:

• Why does it state that the oscillations are rapid unless $|E-E_0|$ is small compared to $\frac{\hbar}{t}$? What is the particular importance of the interval $|E-E_0| \approx \frac{\hbar}{t}$ being narrow compared to $\Delta E$ (width of $|g(E)|^2\rho(E)$) which allows us to assume that $|g(E)|^2 \rho(E)$ is constant (which then implies that $C(t)$ is zero)?

Thanks for any assistance.