How Does Superposition Affect Measurements in a 1-D Harmonic Oscillator?

In summary, the conversation discusses a one-dimensional harmonic oscillator and its wave functions. It also looks at constructing a state for the particle and finding the value of one variable in terms of another. Finally, it explores which particular linear combination will maximize a certain equation.
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Homework Statement
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Relevant Equations
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Consider a one-dimensional harmonic oscillator. ##\psi_0(x)## and ##\psi_1(x)## are the normalized ground state and the first excited states.
\begin{equation}
\psi_0(x)=\Big(\frac{m\omega}{\pi\hbar}\Big)^{\frac{1}{4}}e^{\frac{-m\omega}{2\hbar}x^2}
\end{equation}
\begin{equation}
\psi_1(x)=\Big(\frac{m\omega}{\pi\hbar}\Big)^{\frac{1}{4}}\sqrt{\frac{2m\omega}{\hbar}}xe^{\frac{-m\omega}{2\hbar}x^2}
\end{equation}
(a) Construct a state for the particle that is a linear combination
$$\psi(x)=b_0\psi_0(x)+b_1\psi_1(x)$$
$$\psi(x)=b_0\Big(\frac{m\omega}{\pi\hbar}\Big)^{\frac{1}{4}}e^{\frac{-m\omega}{2\hbar}x^2}+b_1\Big(\frac{m\omega}{\pi\hbar}\Big)^{\frac{1}{4}}\sqrt{\frac{2m\omega}{\hbar}}xe^{\frac{-m\omega}{2\hbar}x^2}$$
Find ##b_1## in terms of ##b_0##.
$$\int_0^a<b_0\psi_0+b_1\psi_1|b_0\psi_0+b_1\psi_1>dx=1$$
$$b_0^2+b_1^2=1$$
$$b_1=\sqrt{1-b_0^2}$$
(b) Which particular linear combination will maximize ##<\psi|\hat{x}|\psi>##?
$$<\psi|\hat{x}|\psi>=\Big<b_0\psi_0(x)+b_1\psi_1(x)\Big|\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}+\hat{a}^{\dagger})\Big|b_0\psi_0(x)+b_1\psi_1(x)\Big>$$
$$=\sqrt{\frac{\hbar}{2m\omega}}\Big<b_0\psi_0(x)+b_1\psi_1(x)\Big|b_1\psi_{0}(x)+b_0\psi_1(x)+b_1\sqrt{2}\psi_2(x)\Big>$$
$$=b_0b_1\sqrt{\frac{\hbar}{2m\omega}}\int^a_0\Big(\psi_0(x)^2+\psi_1(x)^2\Big)dx$$
maximize ##b_0=b_1## ##\rightarrow## ##<\psi|\hat{x}|\psi>##.
$$\frac{d}{db_0}b_0\sqrt{1-b_0^2}=\sqrt{1-b_0^2}-\frac{b_0^2}{\sqrt{1-b_0^2}}=0\Rightarrow b_0,b_1=\sqrt{\frac{1}{2}}$$
$$max(b_0,b_1)\Rightarrow (\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$$
 

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Looks good to me.
 
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Related to How Does Superposition Affect Measurements in a 1-D Harmonic Oscillator?

1. What is a 1-D Harmonic Oscillator?

A 1-D Harmonic Oscillator is a simple physical system that consists of a mass attached to a spring, which is able to move back and forth along a single dimension. It is a common model used in physics to study the behavior of oscillating systems.

2. What is the equation of motion for a 1-D Harmonic Oscillator?

The equation of motion for a 1-D Harmonic Oscillator is given by F = -kx, where F is the restoring force exerted by the spring, k is the spring constant, and x is the displacement of the mass from its equilibrium position.

3. What is the natural frequency of a 1-D Harmonic Oscillator?

The natural frequency of a 1-D Harmonic Oscillator is given by f = 1/(2π√(m/k)), where m is the mass and k is the spring constant. This frequency is a characteristic property of the system and is independent of the amplitude of oscillation.

4. How does the amplitude affect the period of a 1-D Harmonic Oscillator?

The period of a 1-D Harmonic Oscillator is directly proportional to the amplitude of oscillation. This means that as the amplitude increases, the period also increases. However, the natural frequency of the system remains constant regardless of the amplitude.

5. What is the significance of a 1-D Harmonic Oscillator in real-world applications?

1-D Harmonic Oscillators have many real-world applications, such as in clocks, musical instruments, and shock absorbers. They are also used in engineering and physics to model more complex systems and understand their behavior. Additionally, the concept of a 1-D Harmonic Oscillator is fundamental to understanding other types of oscillating systems.

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