Rotate Functions with Derivatives: A Quantum Mechanics Homework

In summary, the conversation discusses the use of the rotation operator in quantum mechanics and its connection to the rotation matrix. The question is raised about how to rotate a function using derivatives and the potential connection to the rotation matrix. The response mentions that momentum is the generator of translation and that infinitesimal translations can be thought of as rotations in the xy plane. It also notes that the rotation operator for finite rotations is different from that for infinitesimal rotations. Finally, there is a question about the rotation of coordinates for a function with complex values.
  • #1
Oliver321
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TL;DR Summary
How is it possible to rotate a function (for example the 3D wave function) by an infinitesimal angle by using derivatives?
I was solving a problem for my quantum mechanics homework, and was therefore browsing in the internet for further information. Then I stumbled upon this here:
F5F459AD-E4C6-4951-A918-DA2E8E80B8D9.jpeg

R is the rotation operator, δφ an infinitesimal angle and Ψ is the wave function.
I know that it is able to rotate a curve, vector... with a rotation matrix. But how is it possible to rotate a function only with derivatives? I tried to rephrase a function f(x) as a curve, applying the 2D rotation matrix and small angle approximation and convert it back to an explicit function f(x). But I did not get the same answer.
My question is now: how does this work and what’s the connection to the rotation matrix?

I am really thankful for every help!
 
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  • #3
I was around the same problem but outside of course material is this correct ?

Suppose ##\psi\in C^\infty(\mathbb{R}^3,\mathbb{C})## then the rotation of coordinates should correspond to a phase : ##e^{i\phi}\psi(\vec{x})=\psi(R\vec{x})\Rightarrow \phi=i(log(\psi(\vec{x})-log(\psi(R\vec{x})))## ?

I asked myself : What about if ##\psi(\vec{x})\in \mathbb{C}^2## ?
 

Related to Rotate Functions with Derivatives: A Quantum Mechanics Homework

1. What is the purpose of rotating functions with derivatives in quantum mechanics homework?

The purpose of rotating functions with derivatives in quantum mechanics homework is to understand the behavior of quantum particles in different orientations or directions. This helps in predicting the probability of finding a particle in a particular state and understanding its overall behavior in a quantum system.

2. How do you rotate a function with derivatives in quantum mechanics?

To rotate a function with derivatives in quantum mechanics, you need to use a mathematical operation called a rotation operator. This operator acts on the wave function of a particle and changes its orientation or direction. The resulting rotated function can then be used to calculate the probability of finding the particle in a particular state.

3. What are the key concepts involved in rotating functions with derivatives in quantum mechanics?

The key concepts involved in rotating functions with derivatives in quantum mechanics include wave functions, rotation operators, and the concept of superposition. Wave functions describe the quantum state of a particle, while rotation operators change the orientation of the wave function. Superposition is the principle that a particle can exist in multiple states simultaneously.

4. Why is it important to understand how to rotate functions with derivatives in quantum mechanics?

It is important to understand how to rotate functions with derivatives in quantum mechanics because it allows us to accurately predict the behavior of quantum particles in different orientations. This is crucial in many real-world applications, such as quantum computing and quantum cryptography, where the behavior of particles in different states is essential to their function.

5. Are there any real-world applications of rotating functions with derivatives in quantum mechanics?

Yes, there are several real-world applications of rotating functions with derivatives in quantum mechanics. Some examples include quantum computing, where the orientation of quantum bits (qubits) is crucial for performing calculations, and quantum cryptography, where the orientation of photons is used for secure communication. Rotating functions with derivatives is also important in understanding the behavior of particles in magnetic fields, which has applications in areas such as medical imaging and materials science.

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