# Tagged Questions

For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales, where the action is on the order of the Planck constant.

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### Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
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### Quantum theory linearly independent solutions

I'm trying to do the part of this qusetion where we need to find two linearly independent solutions to (2) of the given form. Is there a nicer way to do it other than just plugging it into (2). I was ...
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### Quantum oscillator transition amplitude

I have a quick question regarding an equation in my textbook. It's about calculating the probability transition amplitude of a quantum oscillator. Why is this true? The difference between the first ...
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### Non-Geometric Interpretation of the Dot or Inner Product.

I was wondering if there is a non-geometric interpretation of the dot product (or the inner product more generally). That is, an interpretation that has no concept of length and angle. My motivation ...
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### integral of product of three basis functions and Clebsh-Gordan coefficients

Suppose I have an orthonormal basis $\{b_i\}_{i=1}^\infty$ for an $L_2$ space (for example, the $b_i$ could be spherical harmonics on the round sphere with the Euclidean $L_2$ inner product). I want ...
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### Comparing the definition of the Expectation Value to its application in QM

The definition of the expectation value for a continuous domain f(x) is given by $$<f(x)>=\int{f(x)p(x)dx}$$ where p(x) is the probability density function corresponding to {x}. In quantum ...
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### Dirac Notation: What Do $\langle u|v \rangle$ and $\langle u|T|v \rangle$ Represent?

I post this hoping for clarification, and particularly in a context of linear algebra without too much mention of matrices. (1) Let $V$ be a (perhaps infinite dimensional) Hilbert space and $V'$ its ...
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### Singular value decomposition of sum of single particle operators

here is my question. Suppose to have an operator $L$ in a composite Hilbert space $A\otimes B$ which can be written as sum of single particle operator as: L = (L_0\otimes \mathbb{I} + ...
How to prove: $$\int^\pi_0\int^{2\pi}_0 Y_{l'',m''_l}(\theta,\phi) Y_{l',m'_l}(\theta,\phi) Y_{l,m_l}(\theta,\phi) \sin\theta \,d\theta \,d\phi = 0$$ unless $l, l',$ and $l''$ are integers that can ...