I’d like you to write a spaced repetition prompt to reinforce an idea through retrieval practice.
Good prompts should follow these principles:
- Be focused: They should concentrate on one detail at a time to avoid unfocused questions that could hinder recall and concentration.
- Be precise: They need to be explicit about what they're asking to avoid ambiguity and to properly activate targeted knowledge.
- Be consistent: Each recall should aim to 'light the same bulbs', reinforcing the same knowledge each time.
- Be tractable: They should be designed to nearly always yield a correct answer, which might require breaking down the task or adding cues.
- Be effortful: The act of retrieval should involve some level of difficulty. The answer shouldn't be trivial or inferred too quickly. The question should focus retrieval on the reinforcement target without supplying, overconstraining, or unnecessarily cuing the answer.
- Be opinionated: Emphasize rich understanding and important ideas over trivial details.
- Be concise: Questions and answers should be as short as possible.
- Be context-independent: In review, this prompt will be interleaved with many others about many topics. The prompt must cue or supply whatever context is necessary to understand the question.
- Be covert: For whatever knowledge the prompt is trying to reinforce, the question should focus retrieval on that knowledge without supplying, overconstraining, or cuing the answer.
Here is the passage the question will be about.
“Imagine a particle of mass $m$, constrained to move along the $x$ axis, subject to some specified force $F(x, t)$ (Figure 1.1). The program of classical mechanics is to determine the position of the particle at any given time: $x(t)$. Once we know that, we can figure out the velocity $(v=$ $d x / d t)$, the momentum $(p=m v)$, the kinetic energy $\left(T=(1 / 2) m v^2\right)$, or any other dynamical variable of interest. And how do we go about determining $x(t)$ ? We apply Newton's second law: $F=m a$. (For conservative systems - the only kind we shall consider, and, fortunately, the only kind that occur at the microscopic level-the force can be expressed as the derivative of a potential energy function, ${ }^1 F=-\partial V / \partial x$, and Newton's law reads $m d^2 x / d t^2=-\partial V / \partial x$.) This, together with appropriate initial conditions (typically the position and velocity at $t=0$ ), determines $x(t)$
Quantum mechanics approaches this same problem quite differently. In this case what we're looking for is the particle's wave function, $\Psi(x, t)$, and we get it by solving the Schrödinger equation:
$$
i \hbar \frac{\partial \Psi}{\partial t}=-\frac{\hbar^2}{2 m} \frac{\partial^2 \Psi}{\partial x^2}+V \Psi
$$
Here $i$ is the square root of -1 , and $\hbar$ is Planck's constant-or rather, his original constant ( $h$ ) divided by $2 \pi$ :
$$
\hbar=\frac{h}{2 \pi}=1.054573 \times 10^{-34} \mathrm{~J} \mathrm{~s}
$$
The Schrödinger equation plays a role logically analogous to Newton's second law: Given suitable initial conditions (typically, $\Psi(x, 0)$ ), the Schrödinger equation determines $\Psi(x, t)$ for all future time, just as, in classical mechanics, Newton's law determines $x(t)$ for all future time. $^2$”
I would like a prompt focusing on the phrase “The Schrödinger equation plays a role logically analogous to Newton's second law”, specifically emphasizing “how”
Please provide the best possible prompt about this, providing both question and answer. Make sure your prompts follow all the principles above. Mention which of the following desired properties it has: focused, precise, consistent, tractable, effortful, opinionated, concise, context-independent, and covert (just write the words, without explanation). Make sure that you are reinforcing the specific detail I mentioned.
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Prompt: In what way is the role of the Schrödinger equation in quantum mechanics analogous to that of Newton's second law in classical mechanics?
Answer: Both equations, when given suitable initial conditions, determine the state of a system for all future time; Schrödinger equation determines the wave function (Ψ(x, t)), while Newton's second law determines the position (x(t)).
Properties: focused, precise, consistent, tractable, effortful, opinionated, concise, context-independent, covert(notice that the answer is much less focused and concise)
I’d like you to write a spaced repetition prompt to reinforce an idea through retrieval practice.
Here is the passage the question will be about.
“Imagine a particle of mass $m$, constrained to move along the $x$ axis, subject to some specified force $F(x, t)$ (Figure 1.1). The program of classical mechanics is to determine the position of the particle at any given time: $x(t)$. Once we know that, we can figure out the velocity $(v=$ $d x / d t)$, the momentum $(p=m v)$, the kinetic energy $\left(T=(1 / 2) m v^2\right)$, or any other dynamical variable of interest. And how do we go about determining $x(t)$ ? We apply Newton's second law: $F=m a$. (For conservative systems - the only kind we shall consider, and, fortunately, the only kind that occur at the microscopic level-the force can be expressed as the derivative of a potential energy function, ${ }^1 F=-\partial V / \partial x$, and Newton's law reads $m d^2 x / d t^2=-\partial V / \partial x$.) This, together with appropriate initial conditions (typically the position and velocity at $t=0$ ), determines $x(t)$
Quantum mechanics approaches this same problem quite differently. In this case what we're looking for is the particle's wave function, $\Psi(x, t)$, and we get it by solving the Schrödinger equation:
$$
i \hbar \frac{\partial \Psi}{\partial t}=-\frac{\hbar^2}{2 m} \frac{\partial^2 \Psi}{\partial x^2}+V \Psi
$$
Here $i$ is the square root of -1 , and $\hbar$ is Planck's constant-or rather, his original constant ( $h$ ) divided by $2 \pi$ :
$$
\hbar=\frac{h}{2 \pi}=1.054573 \times 10^{-34} \mathrm{~J} \mathrm{~s}
$$
The Schrödinger equation plays a role logically analogous to Newton's second law: Given suitable initial conditions (typically, $\Psi(x, 0)$ ), the Schrödinger equation determines $\Psi(x, t)$ for all future time, just as, in classical mechanics, Newton's law determines $x(t)$ for all future time. $^2$”
I would like a prompt focusing on the phrase “The Schrödinger equation plays a role logically analogous to Newton's second law”, specifically emphasizing “how”
Please provide the best possible prompt about this, providing both question and answer. Make sure that you are reinforcing the specific detail I mentioned.
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Question:
How does the Schrödinger equation play a role logically analogous to Newton's second law in determining the future state of a particle?
Answer:
The Schrödinger equation plays a role logically analogous to Newton's second law because both equations are used to determine the future state of a particle given appropriate initial conditions. In classical mechanics, Newton's second law (F=ma) helps find the position of the particle at any given time (x(t)). Similarly, in quantum mechanics, the Schrödinger equation is used to determine the particle's wave function (Ψ(x, t)) for all future time given suitable initial conditions (typically, Ψ(x, 0)).Using some bits here from a prompt sent to me by Yuval Milo.