All Questions
Tagged with hilbert-spaceeigenvalue
211 questions
2votes
1answer
90views
Analytic approximation of bound state energies for finite square well
Considering a fairly deep finite potential well given by: $$U(x)= \begin{cases} U_0 \ \ , \ \ |x|>\frac{a}{2}\\ 0 \ \ , \ \ |x|\le \frac{a}{2} \end{cases}$$ We know that the energies of the bound ...
1vote
1answer
124views
Can the non-degenerate perturbation theory formula for higher-order energy corrections be used in case of degenerate perturbation theory?
Consider the system given by, $$ H|n\rangle = E|n\rangle$$ where: $H$ is the hamiltonian. $|n\rangle$ is the eigenstate. $E$ is the energy of the eigenstate. Now from $\underline{\textbf{non-...
0votes
1answer
115views
Graphical representations of the vector model of quantum angular momentum
This question is in reference to the book "Introduction to Modern Physics" by Richtmyer and Kennard, particularly in their discussion of the graphical representation of quantized angular ...
1vote
2answers
111views
Trouble understing a passage from Principles of quantum mechanics by Paul Dirac?
I'm currently reading Principles of Quantum Mechanics by Paul Dirac, specifically the 4th edition of 1958. There is a passage I'm having trouble to understand, I'll put the text here. For reference it'...
- 133
2votes
2answers
225views
The eigenvectors associated to the continuous spectrum in Dirac formalism
I am comfused about the definition of an observable, eigenvectors and the spectrum in the physics litterature. All what I did understand from Dirac's monograph is that the state space is a complex ...
- 193
0votes
1answer
63views
Pauli matrix exponentials [closed]
Just a short query to confirm my understanding. Given the Pauli-X operator $\hat{X}$ and it's eigenstates $|+\rangle:=\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|-\rangle:=\frac{1}{\sqrt{2}}(|0\...
- 297
0votes
1answer
97views
What kind of physical process would correspond to an operator that doesn’t result in an eigenvalue equation: $ \hat{A}ψ=a ψ$?
I'm studying quantum mechanics and I'm trying to understand the concept of operators. They can be represented in general by the equation: $$ \hat{A}ψ=ψ'. $$ Here the wavefunction is changed to $ψ'$ ...
- 2,237
2votes
2answers
739views
Change of basis in bra-ket notation [duplicate]
In the post Change of Basis in quantum mechanics using Bra-Ket notation , the accepted answer explores the relationship between an arbitrary operator $\hat{x}$ and another named $\hat{u}$, such that $\...
- 307
0votes
1answer
105views
TISE solutions should be combinations of eigenstates. Why this is not the case? [closed]
I would really appreciate some help with a question I have about the TISE (Sch. tipe independent equation). This is a linear equation and linear combination of the solution should be solution too. The ...
0votes
1answer
129views
An operator with integer eigenvalues?
As is well-known, number operator $N=a^{\dagger}a$ with the commutation relation $[a,a^{\dagger}]=1$ has non-negative integer eigenvalues. I am looking for a similar expression for an operator ($A(a^{\...
- 525
0votes
1answer
145views
Commutable operators and eigenfunctions
My lecture notes says Since $[\hat{L^2} ,\hat{L_{z}}] = 0$ Then $Y(\theta,\phi)$ solves the below equations simultaneously: $$\hat{L^2}Y(\theta,\phi) = A Y(\theta,\phi)$$ $$\hat{L_{z}}Y(\theta,\phi) = ...
- 6,946
0votes
3answers
188views
Why does $Z \otimes Z$ have only two eigenvalues?
The observable $Z = \begin{bmatrix}1 & 0\\\ 0 & -1\end{bmatrix}$ on a single qubit system has two eigenvalues, 1 and -1, which means when measured, the system can give one of two possible ...
- 31
1vote
3answers
3kviews
Physical interpretation of the bra-ket notation
The bra-ket notation generally consists of 'ket', i.e. a vector, and a 'bra', i.e. some linear map that maps a vector to a number in the complex plane. Now, using this bra-ket notation we can compute ...
4votes
2answers
896views
Calculating eigenvalues and eigenstates of an infinite dimensional Hamiltonian
Consider the Hamiltonian, $$H = E_{0} \sum_{m = - \infty}^{\infty}(|m⟩⟨m + 1| + h.c.),$$ where $E_{0}$ is an energy scale, $|m⟩$ are kets which can be used to form a complete basis and h.c. denotes ...
- 322
1vote
0answers
91views
Green Function associated to a periodic Schrodinger operator
If $V:\mathbb R\to \mathbb R$ is an $L$ periodic function in $\operatorname L^{\infty}$ we can always find two independent solutions for $$\psi''(x)+V(x)\psi(x)=E\psi(x)$$ $\psi^{\pm}(x)=e^{\pm ipx}\...
- 147
