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### Is the normalization of the wave-function preserved due to…?

Is the preservation of the inner product the same thing as the vector length of the wave-function staying constant with it's rotation through some R2 plane (ie it's evolution through time), that is ...
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### Inner product of vectors

While reading Shankar Ramamurti's book on Principles of Quantum Mechanics, p. 10, I came across the following lines under the concept of inner product of two vectors (in terms of orthonormal basis): ...
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### Very basic question about quantum field operators

For a matrix $A$, the notation $A^\dagger$ implies the transpose of the complex conjugate of $A$ i.e., $A^\dagger=(A^*)^T$. What does the symbol $\hat{\phi}^\dagger$ mean for a quantum operator ...
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### Is a basis vector always unit-length in a wave function?

I'm currently studying wave functions and I came across an assertion, that $$\psi(x) = \left<x \middle| \psi \right>$$ is a projection of $\psi$ onto $x$. The vector projection being defined ...
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### Proof that if expectation of an operator is zero for all vectors, then the operator itself must be zero

I was attending a Quantum Mechanics lecture when the instructor casually mentioned the following theorem: $\langle \alpha \rvert A \rvert \alpha \rangle = 0 ~\forall \alpha \implies A=0$, where $A$ ...
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### Writing a vector as the sum of basis vectors

I'm currently making my way through quantum mechanics by Leonard Susskind, but have got stuck at this part; writing a vector as the sum of basis vectors. I get that for an $N$ dimensional space and a ...
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### Proof of Schwarz Inequality using Bra-ket notation [closed]

I'm trying to prove Schwarz Inequality, where $$\mid\left\langle \alpha | \beta \right\rangle\mid^2 \leq \left\langle \alpha | \alpha\right\rangle \left\langle \beta| \beta\right\rangle$$ So I ...
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### Rigorous mathematical definition of vector operator?

In standard quantum mechanics textbooks, the concept of operators is often introduced as linear maps that map a Hilbert space $H$ onto itself: $$\hat{O}: H \rightarrow H \, .$$ However, directly ...
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### Basis independence in Quantum Mechanics

The idea that the state of a system does not depend on the basis that we choose to represent it in, has always puzzled me. Physically I can imagine that the basis ought to just yield an equivalent ...
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### What is the meaning of “closure is lost” for a set of kets (or any members of a vector space)?

This is the closure relation in Quantum Mechanics: $$\sum_i |i\rangle \langle i| = 1$$ which I understand as "the sum of the projections onto the basis vectors leaves the projected vector unchanged"...
$(1)$Since quantum-mechanical states between two consecutive measurements are represented as superposition of orthonormal basis vectors in a vector space, at first glance it seems like it makes sense ...