Posing a prompt that gets a useful result currently requires some consideration. Below are some possible tasks for LLMs, along with additional context for the LLM.
Convert Latex to SymPy
How to improve chances of success :
Here's my prompt :
Answer the question based on the context below. Keep the answer short and concise. Respond "Unsure about answer" if not sure about the answer.
Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.
Context: SymPy is an open-source Python library for symbolic computation. SymPy provides a computer algebra system. SymPy can convert Latex math to SymPy, and SymPy can render a mathematical expression as Latex.
Question: What is the SymPy representation for the Latex expression $x^2 + y^2 = 1$ ?
Answer:
Right response : sympy.Eq(x**2 + y**2, 1)
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash" and "Gemini 2.0 Flash Thinking Experimental" each return
"Unsure about answer."
I revised my question to
Latex is software for document preparation to generate
PDF files. Mathematical equations can be expressed in
Latex using markup syntax.
SymPy is an open-source Python library for symbolic
computation. SymPy provides a computer algebra system.
SymPy can convert Latex math to SymPy, and SymPy can
render a mathematical expression as Latex.
Question: What is the SymPy representation for the
Latex expression $x^2 + y^2 = 1$ ?
Answer:
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
1. **Identify the core task:** The question asks for the
SymPy representation of a given LaTeX equation.
2. **Recognize the LaTeX elements:** The LaTeX expression
`$x^2 + y^2 = 1$` contains the following elements:
* Variables: `x`, `y`
* Exponentiation: `^2`
* Addition: `+`
* Equality: `=`
* Constant: `1`
3. **Consider the SymPy equivalents:** Think about how
each LaTeX element maps to SymPy:
* Variables: In SymPy, you need to declare symbols. The
common way to do this is using `Symbol('x')` and `Symbol('y')`.
* Exponentiation: SymPy uses `**` for exponentiation
(like standard Python). So `x^2` becomes `x**2`.
* Addition: Standard Python `+` works for addition in SymPy.
* Equality: A crucial point. In SymPy, the `=` operator is for
assignment. For expressing an *equation*, you need to use
`Eq()`. So `x^2 + y^2 = 1` becomes `Eq(x**2 + y**2, 1)`.
* Constant: The integer `1` is represented as `1` in SymPy.
4. **Assemble the SymPy expression:** Combine the SymPy equivalents:
* First, declare the symbols: `x = Symbol('x')`, `y = Symbol('y')`
* Then, construct the equation using `Eq()`: `Eq(x**2 + y**2, 1)`
5. **Present the answer:** Provide the complete SymPy
code snippet to represent the equation.
6. **Review and refine:** Double-check if the SymPy representation
accurately captures the meaning of the LaTeX expression. Ensure
the use of `Eq()` for equality and the correct syntax for
symbols and exponentiation.
"Gemini 2.0 Flash Thinking Experimental" answer
To find the SymPy representation of the LaTeX expression $x^2 + y^2 = 1$,
you need to translate the mathematical symbols
and structure into their SymPy equivalents.
Here's how you can do it:
Variables: In SymPy, you need to declare symbols before using
them. So, x and y will be represented as Symbol('x') and Symbol('y').
Exponentiation: The ^ in LaTeX corresponds to the ** operator
in Python/SymPy. So, x^2 becomes x**2 and y^2 becomes y**2.
Addition: The + in LaTeX is the same as the + operator in Python/SymPy.
Equality: The = in LaTeX represents mathematical equality.
In SymPy, you represent an equation using the Eq() function.
Therefore, the SymPy representation of $x^2 + y^2 = 1$ is:
from sympy import symbols, Eq
x, y = symbols('x y')
equation = Eq(x**2 + y**2, 1)
content_copy
download
Use code with caution.
Python
So the answer is:
Eq(x**2 + y**2, 1)
Next prompt to explore :
Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.
Context: SymPy is an open-source Python library for symbolic computation. SymPy provides a computer algebra system. SymPy can convert Latex math to SymPy, and SymPy can render a mathematical expression as Latex.
Question: What is the SymPy representation for the Latex expression $\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}$ ?
Answer:
Right response :Eq(p_electron, p1 - p2)
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
Let's break down the thought process for converting the given LaTeX expression to its SymPy representation.
1. **Identify the core components of the LaTeX expression:** The expression is $\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}$. This involves:
* Vectors: $\vec{p}_{electron}$, $\vec{p}_{1}$, $\vec{p}_{2}$
* Subscripts: `electron`, `1`, `2`
* Equality: `=`
* Subtraction: `-`
2. **Consider how SymPy handles different mathematical entities:**
* **Variables:** In SymPy, we need to declare symbols. The base names are `p`.
* **Vectors:** SymPy has a `vector` module. We can represent vectors using `CoordSys3D` and vector components, or simply as vector symbols. For simpler representations, we might just treat the vectors as symbols themselves, making it easier to work with them symbolically without focusing on their components. Since the context doesn't specify dimensionality or require component-wise operations, treating the vectors as distinct symbolic entities seems like a good starting point.
* **Subscripts:** LaTeX subscripts are often used for indexing or distinguishing variables. In SymPy, we can't directly have subscripts in variable names. We can handle this in a few ways:
* Use separate variable names: `p_electron`, `p1`, `p2`. This is the most straightforward and Pythonic approach.
* Use indexed symbols (from `sympy.IndexedBase` and `sympy.Idx`). This is more suitable if you are working with sequences or arrays of variables, and you need to manipulate the indices. For this specific problem, it's likely overkill.
* **Equality:** The `=` in LaTeX translates to the `Eq()` function in SymPy to represent an equation.
* **Subtraction:** The `-` operator in LaTeX is the same as the `-` operator in SymPy.
3. **Formulate the SymPy representation:** Based on the above analysis, a natural way to represent the expression is to use distinct symbols for each vector:
* `\vec{p}_{electron}` -> `p_electron`
* `\vec{p}_{1}` -> `p1`
* `\vec{p}_{2}` -> `p2`
Now, assemble the equation using `Eq()`:
`Eq(p_electron, p1 - p2)`
4. **Include necessary imports:** To use `symbols` and `Eq`, we need to import them from the `sympy` library.
`from sympy import symbols, Eq`
5. **Declare the symbols:** Before using the symbols, they need to be declared using `symbols()`.
`p_electron, p1, p2 = symbols('p_electron p1 p2')`
6. **Combine the parts:** Put everything together to form the complete SymPy code:
```python
from sympy import symbols, Eq
p_electron, p1, p2 = symbols('p_electron p1 p2')
equation = Eq(p_electron, p1 - p2)
```
7. **Consider alternative representations (and why the chosen one is suitable):**
* **Using `IndexedBase`:** We could represent the vectors using indexed symbols, like `p = IndexedBase('p')`, and then use `p['electron']`, `p[1]`, `p[2]`. However, the subscript "electron" isn't a numerical index, making this less direct. While possible, it adds complexity for this specific problem.
* **Using explicit vector objects:** SymPy's `vector` module allows defining coordinate systems and vectors. This would be necessary if we were performing vector operations like dot products or cross products and needed to work with components. However, the given equation is a simple symbolic equation between vectors, so treating them as abstract symbols is sufficient.
8. **Final check:** The SymPy representation `Eq(p_electron, p1 - p2)` accurately reflects the mathematical structure of the LaTeX expression. It uses appropriate SymPy constructs for equality, subtraction, and symbolic variables.
"Gemini 2.0 Flash Thinking Experimental" answer
To represent the LaTeX expression $\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}$ in SymPy, we need to consider how to represent vectors and subscripts. Since we are dealing with symbolic manipulation and not necessarily with the components of the vectors, we can treat the vectors as distinct symbolic entities.
Here's how you can represent it in SymPy:
```python
from sympy import symbols, Eq
# Represent the vectors as symbolic variables
p_electron, p1, p2 = symbols('p_electron p1 p2')
# Form the SymPy equation
equation = Eq(p_electron, p1 - p2)
```
Therefore, the SymPy representation is:
```python
Eq(p_electron, p1 - p2)
```
**Explanation:**
* **`symbols('p_electron p1 p2')`**: We use `symbols` to create
symbolic variables representing the vectors. While LaTeX
uses $\vec{p}$ notation and subscripts, in SymPy, we can use
more descriptive variable names like `p_electron`, `p1`, and `p2`.
This maintains the distinction between the different momentum vectors.
* **`Eq(p_electron, p1 - p2)`**: The `Eq()` function from SymPy
is used to represent an equation. The first argument is the left-hand
side, and the second argument is the right-hand side. The
subtraction `-` operator works as expected for symbolic variables.
**Alternative (Less Common for this simple case):**
You could potentially use `IndexedBase` if you were dealing with
a more general set of vectors and wanted to use numerical indices:
```python
from sympy import symbols, Eq, IndexedBase, Idx
p = IndexedBase('p')
i, j = symbols('i j', integer=True) # Not really needed here but for demonstration
electron = symbols('electron') # Could use a string or symbol for 'electron'
# This becomes less direct as 'electron' isn't a typical index
# equation = Eq(p[electron], p[1] - p[2])
```
However, for this specific case, representing each vector with a
distinct symbol like `p_electron`, `p1`, and `p2` is the most
straightforward and readable way to represent the equation in SymPy.
Next prompt to explore :
Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.
Context: SymPy is an open-source Python library for symbolic computation. SymPy provides a computer algebra system. SymPy can convert Latex math to SymPy, and SymPy can render a mathematical expression as Latex.
Question: What is the SymPy representation for the Latex expression $x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$ ?
Answer:
Right response :
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
Here's a breakdown of my thought process to arrive at the SymPy representation of the given LaTeX expression:
1. **Identify the Components:** The first step is to break down the LaTeX expression into its constituent parts and their mathematical meaning:
* `x`: A single variable.
* `=`: Equality.
* `\langle ... | ... \rangle`: The Dirac bra-ket notation for the inner product (or expectation value in this context).
* `\psi_{\alpha}`: A ket vector labeled by $\alpha$.
* `\hat{A}`: An operator.
* `\psi_{\beta}`: A bra vector labeled by $\beta$.
2. **Consider SymPy Equivalents:** Now, think about how each of these components can be represented in SymPy:
* **Variables:** Simple variables like `x` can be represented using `Symbol('x')`.
* **Equality:** The `=` is represented by `Eq()`.
* **Bra-Ket Notation:** SymPy has specific modules for quantum mechanics, including bra-ket notation. The key is the `sympy.physics.quantum.state` module, which provides `Ket` and `Bra` classes, and `sympy.physics.quantum.operator` for operators.
* **Vectors/States:** The ket `|\psi_{\beta}\rangle` can be represented using `Ket('psi', label='beta')`. Similarly, the bra `\langle\psi_{\alpha}|` can be represented using `Bra('psi', label='alpha')`. The label helps distinguish between different states.
* **Operators:** The operator `\hat{A}` can be represented using `Operator('A')`.
* **Inner Product:** The inner product $\langle \phi | \hat{O} | \psi \rangle$ is conceptually a multiplication of the bra, operator, and ket. In SymPy's quantum framework, this is represented by simply multiplying these objects.
3. **Construct the SymPy Expression:** Based on the above mapping, build the SymPy code:
* Import necessary modules: `from sympy import symbols`, `from sympy.physics.quantum.state import Ket, Bra`, `from sympy.physics.quantum.operator import Operator`.
* Define the symbols: `x = symbols('x')`.
* Define the quantum objects: `psi_alpha_bra = Bra('psi', label='alpha')`, `A_hat = Operator('A')`, `psi_beta_ket = Ket('psi', label='beta')`.
* Form the inner product: `inner_product = psi_alpha_bra * A_hat * psi_beta_ket`.
* Create the equation: `equation = Eq(x, inner_product)`.
4. **Refine and Verify:** Review the constructed SymPy expression to ensure it accurately reflects the original LaTeX. The order of the bra, operator, and ket is crucial in the inner product.
5. **Consider Alternatives (and why they are less suitable here):**
* **Direct String Representation:** One might initially think of just using strings, but SymPy won't understand the mathematical relationships and won't be able to perform symbolic manipulations.
* **Generic Symbols:** While you *could* represent the bra, operator, and ket as generic symbols, you lose the specific meaning and the capabilities of SymPy's quantum framework. Using `Ket`, `Bra`, and `Operator` allows SymPy to understand the context and apply relevant rules.
* **IndexedBase (Less appropriate here):** `IndexedBase` is useful for sequences or arrays of symbols, but here we have distinct quantum objects, so the dedicated quantum classes are more appropriate.
By following this structured approach, breaking down the LaTeX, mapping to SymPy concepts, and then assembling the expression, I can arrive at the correct and meaningful SymPy representation.
"Gemini 2.0 Flash Thinking Experimental" answer
To represent the LaTeX expression $x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$ in SymPy, you need to represent the variables, the bra-ket notation, and the operator. SymPy's `sympy.physics.quantum` module provides tools for this.
Here's how you can represent it:
```python
from sympy import symbols, Eq
from sympy.physics.quantum.state import Bra, Ket
from sympy.physics.quantum.operator import Operator
# Define the symbols
x = symbols('x')
# Define the quantum states
psi_alpha = Bra('psi', label='alpha')
psi_beta = Ket('psi', label='beta')
# Define the operator
A_hat = Operator('A')
# Construct the inner product
inner_product = psi_alpha * A_hat * psi_beta
# Form the equation
equation = Eq(x, inner_product)
```
Therefore, the SymPy representation is:
```python
Eq(x, Bra('psi', label='alpha') * Operator('A') * Ket('psi', label='beta'))
```
**Explanation:**
* **`from sympy import symbols, Eq`**: Imports the necessary functions for creating symbols and equations.
* **`from sympy.physics.quantum.state import Bra, Ket`**: Imports the `Bra` and `Ket` classes for representing quantum states.
* `Bra('psi', label='alpha')` represents $\langle\psi_{\alpha}|$. 'psi' is the label of the state, and 'alpha' is a parameter or index distinguishing this particular state.
* `Ket('psi', label='beta')` represents $|\psi_{\beta}\rangle$.
* **`from sympy.physics.quantum.operator import Operator`**: Imports the `Operator` class for representing quantum operators.
* `Operator('A')` represents $\hat{A}$.
* **`x = symbols('x')`**: Defines the variable `x`.
* **`inner_product = psi_alpha * A_hat * psi_beta`**: Represents the inner product $\langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$. SymPy's quantum module overloads the multiplication operator `*` to represent the application of operators and the formation of inner products in the correct order.
* **`equation = Eq(x, inner_product)`**: Forms the SymPy `Eq` object representing the equation $x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$.
This representation allows you to perform symbolic manipulations on this quantum mechanical expression using SymPy's capabilities.
Specify the mathematical relation between period and frequency as an equation in Latex
How to improve chances of success :
Caveat : the page https://en.wikipedia.org/wiki/Frequency includes the statement that
"period is the reciprocal of the frequency: f = 1/T."
Use the context below to produce a result. Keep the response short and concise. Respond "Unsure about answer" if not sure about the answer.
Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.
Context: Period (symbol T) is the interval of time between events. Frequency (symbol f) is the number of occurrences of a repeating event per unit of time
Specify the mathematical relation between period and frequency as an equation in Latex.
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
"Gemini 2.0 Flash Thinking Experimental" answer
Find arxiv papers with derivations
How to improve chances of success :
Explain arxiv
define what I mean by a derivation
Provide example citations
Provide citations based on the context below.
Context: arxiv is an open-access repository of electronic preprints
Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.
Cite three papers from arxiv that contain mathematical derivations with more than four steps.
Identify derivation steps between physics equations
How to improve chances of success :
define what I mean by a derivation
Provide example steps
Answer the question based on the context below. Keep the answer short.
Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.
Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.
Question: What mathematical steps relate the Latex math expression $i x = log(y)$ and $\exp(i x) = y$ ?
Answer:
Right answer : Raise both sides as the power of $\exp$
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
The user is asking for the mathematical step that transforms the
equation $ix = \log(y)$ into $\exp(ix) = y$. This involves the
inverse operation of the logarithm, which is the exponential function.
"Gemini 2.0 Flash Thinking Experimental" answer
Exponentiation.
Answer the question based on the context below. Keep the answer concise.
Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.
Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.
Question: What is the derivative of $y = \cos(x) + i \sin(x)$ with respect to $x$ ?
Answer:
Right answer : $\frac{d}{dx} y = -\sin(x) + i\cos(x)$
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
The question asks for the derivative of a given expression. I need to apply the rules of differentiation.
* The expression is a sum of two terms.
* The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$.
* The derivative of $i \sin(x)$ with respect to $x$ is $i \cos(x)$ (since $i$ is a constant). "Gemini 2.0 Flash Thinking Experimental" answer
$-\sin(x) + i \cos(x)$ Derive the wave function for a quantum particle in a 1D box I removed "Keep the answer concise. Respond "Unsure about answer" if not sure about the answer. Let's work this out in a step by step way to be sure we have the right answer." from the prompt.
Answer the question based on the context below. Express your answer using Latex for the math. Describe the transition between each step.
Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.
Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.
Question: derive the wave function for a quantum particle in a 1D box
Answer:
Right answer: see https://derivationmap.net/review_derivation/000010/
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
The user wants a Latex-formatted derivation of the wave function for a particle in a 1D box. I need to break down the derivation into logical steps and provide the Latex code for each step. Each step must be accompanied by a description of the transition from the previous step.
Here is a potential breakdown of the steps:
1. **Start with the time-independent Schrodinger equation.**
2. **Apply it to the case of a 1D box (potential definition).**
3. **Solve the differential equation in the region within the box.**
4. **Apply boundary conditions.**
5. **Normalize the wave function.**
For each step, I'll need to explain the physical/mathematical reasoning behind the transformation. "Gemini 2.0 Flash Thinking Experimental" answer
The derivation of the wave function for a quantum particle in a 1D box proceeds as follows:
**Step 1: Start with the time-independent Schrödinger equation.**
The time-independent Schrödinger equation describes the stationary states of a quantum system. It is given by:
```latex
-\frac{\hbar^2}{2m} \frac{d^2 \Psi(x)}{dx^2} + V(x) \Psi(x) = E \Psi(x)
```
*Transition:* We begin with the fundamental equation governing the behavior of quantum particles in stationary states.
**Step 2: Define the potential for a 1D box.**
For a 1D box of length $L$, the potential $V(x)$ is zero inside the box ($0 < x < L$) and infinite outside the box ($x \leq 0$ or $x \geq L$).
```latex
V(x) = \begin{cases} 0 & 0 < x < L \\ \infty & \text{otherwise} \end{cases}
```
Inside the box, where $V(x) = 0$, the Schrödinger equation simplifies to:
```latex
-\frac{\hbar^2}{2m} \frac{d^2 \Psi(x)}{dx^2} = E \Psi(x)
```
*Transition:* We specialize the Schrödinger equation to the specific scenario of a particle confined within a 1D box by defining the potential energy function. Since the potential is infinite outside the box, the probability of finding the particle there is zero, meaning $\Psi(x) = 0$ for $x \leq 0$ and $x \geq L$.
**Step 3: Solve the differential equation inside the box.**
Rearranging the simplified Schrödinger equation, we get:
```latex
\frac{d^2 \Psi(x)}{dx^2} = -\frac{2mE}{\hbar^2} \Psi(x)
```
Let $k^2 = \frac{2mE}{\hbar^2}$. Then the equation becomes:
```latex
\frac{d^2 \Psi(x)}{dx^2} = -k^2 \Psi(x)
```
The general solution to this second-order linear homogeneous differential equation is:
```latex
\Psi(x) = A \sin(kx) + B \cos(kx)
```
where $A$ and $B$ are arbitrary constants.
*Transition:* We solve the simplified Schrödinger equation within the boundaries of the box, obtaining a general solution involving sine and cosine functions.
**Step 4: Apply boundary conditions.**
Since the potential is infinite outside the box, the wave function must be zero at the boundaries of the box: $\Psi(0) = 0$ and $\Psi(L) = 0$.
Applying the first boundary condition, $\Psi(0) = 0$:
```latex
\Psi(0) = A \sin(k \cdot 0) + B \cos(k \cdot 0) = A \cdot 0 + B \cdot 1 = B
```
Therefore, $B = 0$. The wave function now becomes:
```latex
\Psi(x) = A \sin(kx)
```
Applying the second boundary condition, $\Psi(L) = 0$:
```latex
\Psi(L) = A \sin(kL) = 0
```
For a non-trivial solution (where $A \neq 0$), we must have:
```latex
\sin(kL) = 0
```
This implies that $kL = n\pi$, where $n$ is an integer ($n = 1, 2, 3, ...$). Therefore, the allowed values of $k$ are:
```latex
k_n = \frac{n\pi}{L}
```
Since $k^2 = \frac{2mE}{\hbar^2}$, the allowed energy levels are:
```latex
E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{\hbar^2}{2m} \left(\frac{n\pi}{L}\right)^2 = \frac{n^2 \pi^2 \hbar^2}{2mL^2}
```
The wave function becomes:
```latex
\Psi_n(x) = A \sin\left(\frac{n\pi x}{L}\right)
```
*Transition:* We enforce the physical constraint that the particle cannot exist outside the box by applying boundary conditions to the general solution. This quantization leads to discrete allowed values for the wave number $k$ and consequently, for the energy $E$.
**Step 5: Normalize the wave function.**
The probability of finding the particle somewhere within the box must be 1. This is expressed by the normalization condition:
```latex
\int_{0}^{L} |\Psi_n(x)|^2 dx = 1
```
Substituting the wave function:
```latex
\int_{0}^{L} \left|A \sin\left(\frac{n\pi x}{L}\right)\right|^2 dx = 1
```
Assuming $A$ is real, we have:
```latex
A^2 \int_{0}^{L} \sin^2\left(\frac{n\pi x}{L}\right) dx = 1
```
Using the identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$:
```latex
A^2 \int_{0}^{L} \frac{1 - \cos\left(\frac{2n\pi x}{L}\right)}{2} dx = 1
```
Integrating, we get:
```latex
A^2 \left[ \frac{x}{2} - \frac{L}{4n\pi} \sin\left(\frac{2n\pi x}{L}\right) \right]_{0}^{L} = 1
```
```latex
A^2 \left( \frac{L}{2} - 0 \right) = 1
```
```latex
A^2 \frac{L}{2} = 1
```
Solving for $A$:
```latex
A = \sqrt{\frac{2}{L}}
```
Therefore, the normalized wave function for a quantum particle in a 1D box is:
```latex
\Psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)
```
for $0 < x < L$, and $\Psi_n(x) = 0$ otherwise, where $n = 1, 2, 3, ...$.
*Transition:* We ensure that the wave function represents a physically realizable probability distribution by normalizing it, determining the specific value of the constant $A$. This completes the derivation of the wave function for a particle in a 1D box.
Derive Newton's Law of Universal Gravitation
Answer the question based on the context below. Keep the answer short and concise. Respond "Unsure about answer" if not sure about the answer. Let's work this out in a step by step way to be sure we have the right answer.
Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.
provide a derivation of Newton's Law of Universal Gravitation
See https://physicsderivationgraph.blogspot.com/2023/06/finding-derivations-of-newtons-law-of.html
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
"Gemini 2.0 Flash Thinking Experimental" answer
Convert derivation steps to a proof in Lean
How to improve chances of success :
define what I mean by a derivation
Explain Lean
Provide example
Emphasize correctness and precision
I removed "Keep the answer short and concise. Respond "Unsure about answer" if not sure about the answer. Let's work this out in a step by step way to be sure we have the right answer." from the prompt.
Answer the question based on the context below.
Context: Lean 4 is a proof assistant and programming language. Lean is based on the calculus of constructions with inductive types.
`a=b` is a proposition. The variable `a` is Real valued and `b` is Real valued. Prove in Lean that `(a=b) -> (a+2=b+2)`
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
Let's break down the thought process for constructing that Lean 4 proof.
1. **Understanding the Goal:** The fundamental goal is to prove the implication `(a = b) -> (a + 2 = b + 2)` within the Lean 4 environment. This means we need to show that *if* `a` is equal to `b`, *then* `a + 2` is equal to `b + 2`.
2. **Identifying the Tool: Rewriting:** The core idea to manipulate equalities in proof assistants like Lean is *rewriting*. If we know `a = b`, we can substitute `b` for `a` (or `a` for `b`) within an expression. This seems like the most direct approach here.
3. **Translating to Lean Syntax (Initial Thoughts):**
- We need to introduce the hypothesis `a = b`. In Lean, this is done using `assume` or `intro`. Let's use `intro h`.
- We need to show `a + 2 = b + 2`.
- How do we apply the equality `h`? Lean has tactics like `rewrite` or `rw`.
4. **First Attempt (Mental Simulation):**
```lean
theorem my_theorem (a b : ℝ) : (a = b) -> (a + 2 = b + 2) := by
intro h
-- We have h : a = b. We want to show a + 2 = b + 2
-- Maybe rewrite h?
rw [h] -- This will replace 'a' with 'b' in the goal.
-- The goal will become b + 2 = b + 2, which is trivially true.
exact rfl
```
This seems promising. The `rw [h]` tactic looks like it will do the substitution.
5. **Refining and Formalizing the Lean Code:**
- **Declare variables:** We need to tell Lean that `a` and `b` are real numbers. `variables (a b : ℝ)` does this correctly.
- **Define the theorem:** We need to give the theorem a name and state its type (the implication). `theorem my_theorem (a b : ℝ) : (a = b) -> (a + 2 = b + 2) := by` is the standard way to start.
- **Introduce the hypothesis:** `intro h` introduces the assumption `a = b` and names it `h`.
- **Apply the rewrite:** `rw [h]` performs the substitution. Lean is smart enough to know to substitute `b` for `a` because the goal has `a + 2` on the left.
- **Prove the reflexivity:** After the rewrite, the goal becomes `b + 2 = b + 2`. This is a fundamental property of equality, known as reflexivity. Lean provides the `rfl` tactic (reflexivity) to prove such goals.
6. **Considering Alternatives (and why they might be less direct):**
- **Equational Reasoning with `calc`:** We *could* use a more verbose `calc` block:
```lean
theorem my_theorem (a b : ℝ) : (a = b) -> (a + 2 = b + 2) := by
intro h
calc
a + 2 = b + 2 := by rw [h]
```
This works but is a bit more verbose for this simple case. `rw` is more direct.
- **Using `subst`:** The `subst` tactic is another way to handle equalities. `subst h` would also replace `a` with `b` in the goal. It's functionally similar to `rw` in this scenario.
7. **Final Check and Explanation:** Review the code to ensure it's clear, concise, and correct. Provide a step-by-step explanation of what each part of the Lean code does and how it proves the theorem. Emphasize the key concepts like introducing hypotheses and using rewriting.
This systematic approach, moving from understanding the problem to identifying appropriate tactics and then formalizing the Lean code, is a common strategy for proving theorems in proof assistants. The mental simulation of how the tactics will affect the proof state is crucial.
"Gemini 2.0 Flash Thinking Experimental" answer
```lean
variables (a b : ℝ)
theorem my_theorem : (a = b) -> (a + 2 = b + 2) := by
intro h,
rw [h],
exact rfl
```
**Explanation:**
1. **`variables (a b : ℝ)`**: This line declares that `a` and `b` are variables of type `ℝ`, which represents the real numbers in Lean. This matches the context stating they are "Real valued".
2. **`theorem my_theorem : (a = b) -> (a + 2 = b + 2) := by`**: This line starts the definition of a theorem named `my_theorem`. The type of the theorem is `(a = b) -> (a + 2 = b + 2)`, which is the implication we want to prove. The `by` keyword starts the proof script.
3. **`intro h,`**: This tactic introduces the hypothesis `a = b` and gives it the name `h`. This is the standard way to handle an implication (`->`) in Lean. We assume the left-hand side of the implication is true.
4. **`rw [h],`**: This is the core of the proof. The `rw` tactic (short for "rewrite") is used to replace occurrences of the left-hand side of an equality with its right-hand side (or vice-versa). In this case, `h` is the hypothesis `a = b`. The `rw [h]` command looks at the current goal, which is `a + 2 = b + 2`, and finds an occurrence of `a` on the left-hand side. It then replaces this `a` with `b` based on the equality `h`. Therefore, the goal becomes `b + 2 = b + 2`.
5. **`exact rfl`**: After the `rw` tactic, the goal is now `b + 2 = b + 2`. This is a trivial equality; something is always equal to itself. The `rfl` tactic (short for "reflexivity") proves such goals. It states that the left-hand side is definitionally equal to the right-hand side.
**In summary, the proof works by assuming `a = b` and then using this equality to rewrite the expression `a + 2` to `b + 2`, thus proving `a + 2 = b + 2`.**
I then clarified
Please use ASCII rather than Unicode to write the Lean 4 syntax
and Gemini provided
```lean
variables (a b : Real)
theorem my_theorem : (a = b) -> (a + 2 = b + 2) := by
intro h,
rw [h],
exact rfl
```
**Explanation of ASCII version:**
The only change needed is replacing the Unicode symbol `ℝ` with its ASCII equivalent in Lean, which is `Real`.
* **`variables (a b : Real)`**: This declares `a` and `b` as variables of type `Real`.
* The rest of the syntax remains the same as it already uses ASCII characters.
This ASCII version is functionally identical to the original Unicode version in Lean 4.
I tried compiling with Lean and learned the syntax was incorrect. The following is closer to correct
import Mathlib.Data.Real.Basic
variable (a b : Real)
example : (a = b) -> (a + 2 = b + 2) := by
intro h
rw [h]
exact rfl
but has the error "no goals to be solved"
Identify symbols in latex arxiv papers
How to improve chances of success :
Provide example
Emphasize correctness and precision
Answer the question based on the context below. Keep the answer short and concise. Respond "Unsure about answer" if not sure about the answer.
Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.
Question: What mathematical expressions are present in the following Latex?
```
\begin{equation}
a = b + c
\end{equation}
Assuming you know $b$, then
\begin{equation}
b*c = f
\end{equation}
```
Answer:
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
"Gemini 2.0 Flash Thinking Experimental" answer
Answer the question based on the context below. Keep the answer short and concise.
Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.
Question: What variables are present in the following Latex?
```
\begin{equation}
a = b + c
\end{equation}
Assuming you know $b$, then
\begin{equation}
b*c = f
\end{equation}
```
Answer:
As of 2025-01-25 on https://aistudio.google.com/ , "Gemini 2.0 Flash Thinking Experimental" returns the following:
Thoughts
"Gemini 2.0 Flash Thinking Experimental" answer
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