# Questions tagged [normalization]

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### Normalization for a free Dirac plane wave

I've recently come about the free plane wave solutions to the Dirac equation, and i'm having a hard time proving that the normalization factor $n$ is $$n=\frac{1}{\sqrt{2m(m+\omega)}}$$ Where $\omega$ ...
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### Confusion about ket states and bra with position

I am very confused about the bra-ket notation of states and the fact that $$\psi(x) = ⟨x|\psi⟩$$ and $$⟨x|x'⟩ = \delta(x-x')$$ are true. What does this mean? What is the ket $|x⟩$, is it just some ...
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### Dirac Delta Function and Position [duplicate]

How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator $\hat{x}$? In math, why does $\langle x’|x\rangle = \delta(x’-x)$?
If I Fourier transform a wave function in position space, integration carries no normalization constant: $$\displaystyle{\phi(k) \equiv \langle k|\psi\rangle = \sum\limits_x \langle k|x\rangle\langle ... 1answer 106 views ### Orthonormality and completeness in infinite dimensions: 2 different definitions [duplicate] In finite dimensional vector spaces, orthonormality is defined as \langle x_i|x_j \rangle=\delta_{ij} and the completeness relation is given simply by$$I = \sum_i |x_i\rangle\langle x_i|.$$To me, ... 0answers 85 views ### Orthonormality: from finite (\delta_{ij}) to infinite (\delta(x-y)) dimensional vector spaces [duplicate] I've been reading Shankar's book on QM, but I'm unsatisfied with the section on "Generalization to Infinite Dimensions". Given a finite dimensional vector space with a basis \{x_i\}, I understand ... 2answers 186 views ### Linear combination of 2 spherical harmonic functions The task is to form 2 linear combinations out of the 2 given spherical harmonic functions. I dont understand why the resultant wave function has to be multiplied with the constant 1/sqrt(2)? 1answer 257 views ### Need for normalization in non-degenerate perturbation theory I'm currently taking a class in QM and we came across the topic of non-degenerate perturbation theory. Let us for further discussions assume that H_0 is the unperturbed Hamiltonian with solutions of ... 1answer 147 views ### Ladder operators and energy levels [closed] I am studying how to get the normalization factor algebraically in the exited states of the harmonic oscillator using the beautiful ladder operators technique. I am stuck at the point where is stated ... 0answers 65 views ### Why are periodic functions through space disregarded in quantum mechanics? My question is a general question arising from an observation from the wavefunction of a free particle. One must disregard, for a free particle, that the spatial wavefunction is:$$\psi(x) = \sum_n ...
One can show that a possible solution to a wavefunction with constantly zero potential is equal to, only considering the spacial piece: $$\psi(x) = \int_{-\infty}^{\infty} A(k) \ e^{ikx} dk$$ This ...