All Questions
Tagged with linear-algebrawavefunction
26 questions
0votes
0answers
66views
Probability of measuring $S_z$ when the particle has an arbitrary spin direction
Suppose we have to find the eigen values of spin operator $\hat{S}$ along a unit vector $\hat{n}$ that lies in XZ plane and then we are to find the probability measuring $S_z$ with $+\frac{\hbar}{2}$ ...
0votes
1answer
121views
Operators algebra for quantum mechanics [closed]
I am taking my first quantum mechanics course and I am a bit lost in operators algebra. These are the main questions I have: Why can we write this kind of equations? $$ Ô \psi = o\psi $$ What I mean ...
0votes
2answers
291views
Is it necessary that the Hilbert space basis is made up of the eigenvectors/eigenfunctions of the operator under consideration?
When using Dirac bra-ket notation to make some statement regarding an operator acting on the vectors of a Hilbert space, is it necessary that the basis of the Hilbert space is made up of the ...
- 153
0votes
0answers
142views
What's the relationship between energy level and position in quantum mechanics?
In my QM class, my professor is having us do a project on numerically solving the Schrödinger equation for a particle in a square well based on this tutorial. I understand the tutorial fine, but one ...
0votes
2answers
440views
Why do most introductory texts on QM use the Schrödinger formulation rather than Heisenberg's matrix mechanics? [closed]
I know they're mathematically equivalent, and that makes intuitive sense, seeing as linear differential equations can in general be solved using matrices and other linear algebra approaches. In fact, ...
2votes
1answer
432views
Quantum state in continuous basis [duplicate]
If I have an arbitrary state $|\psi\rangle$ and want to represent it in a continuous basis, for example the position basis in $x$-direction, I will get $$|\psi\rangle = \int dx\, \langle x|\psi\rangle|...
4votes
1answer
195views
How can the position representation make sense with compatibility of addition? (Dirac Notation)
According to the definition of complex inner product is that: $$⟨\psi|\phi_{1} + \phi_{2}⟩ = \left<\psi|\phi_{1}\right> + \left< \psi| \phi_{2} \right>, \forall \psi, \phi.$$ This implies ...
0votes
2answers
202views
Why is the Schrödinger Equation valid for the component functions (wave function) of state vectors?
I'm new to quantum mechanics and confused about the way the Schrödinger equation is used (more general eigenvalue equations of observables). Let's take the time-independent Schrödinger equation (...
0votes
1answer
125views
Energy (Hamiltonian) of Trial Wavefunction
Here I give a part of derivation of Hartree-Fock equations in case where basis functions (wavefunctions) are orthonormal and real: $$ \langle \psi_i | \psi_j \rangle = \langle \psi_j | \psi_i \rangle =...
- 615
0votes
1answer
80views
Rewriting $|\Psi\rangle=\sum_n c_n |\Phi_n\rangle$ into $|\Phi_n\rangle$ as a function of $c_n$ and $|\Psi\rangle$
Given that $\{|\Phi_n\rangle\}$ is an orthonormal basis, how can I express $|\Phi_n\rangle$ in $c_n$ and $|\Psi\rangle$? \begin{equation} |\Psi\rangle=\sum_n c_n |\Phi_n\rangle \end{equation}
0votes
2answers
786views
The general wavefunction can be expanded in such eigenstates
Suppose we have solved for the energy eigenstates of some Hamiltonian operator $\hat{H}$. We call the energy eigenstates $\psi_n (x)$, where: $n=1$: $\psi_1 (x)$ is the ground states $n=2$: ...
- 187
0votes
1answer
897views
Quantum Harmonic Oscillator and Diagonalization
Suppose we want to find the eigenvalues and the eigenfunctions of the following 3D Hamiltonian: $$H=\frac{p_x^2+p_x^2+p_y^2}{2m}+\frac{1}{2}m \omega ^2(2x^2+2y^2+2xy+z^2)$$ Now: On my own, right now, ...
- 4,683
0votes
1answer
313views
Matrix elements of operators in position representation
In a lecture note, it is written $$ T_{ij} = \langle \phi_i| \hat{T} | \phi_j \rangle = \int d^3 \vec{r} \phi_i^*(\vec{r}) T(\vec{r}) \phi_j(\vec{r}) $$ How to obtain the second integral form from ...
- 49
0votes
1answer
307views
Doubt in a solved example from Quantum Mechanics: Concepts and Applications by Nouredine Zettili [closed]
Question 3.7 b) from Quantum Mechanics: Concepts and Applications by Nouredine Zettili, on page no. 188 (solved examples) - I understand all the solutions mentioned therein but can't figure out why ...
- 435
3votes
3answers
601views
Why should bras be thought of as linear functionals?
Quoting from Ballentine's textbook on Quantum Mechanics: There are situations in which it is important to remember that the primary definition of the bra vector is as a linear functional on the space ...
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