# Questions tagged [linear-algebra]

To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.

525 questions
Filter by
Sorted by
Tagged with
46 views

41 views

### Orthonormalization of eigenamplitudes

Assuming $(-\omega^2 \hat m + \hat k)\vec{a}=0$ where $\vec a$ is the eigenamplitude of the eigenfrequency $\omega$ , $\hat m$ is the mass matrix and $\hat k$ is the matrix of the potential constants. ...
160 views

### Why must momentum operator in infinite well be self adjoint?

First, let me preface this statement by saying I know that there exists no (unique) self adjoint extension of the standard differential operator for the space $L_2([0,1])$. However, when one attempts ...
112 views

### What are the Eigenstates and Eigenvalues? [closed]

In quantum mechanics I keep hearing about them. Kindly tell about them...not at a very very high level but simple enough to understand completely
37 views

48 views

### Simultaneous shifted diagonalization of bunch of operators

I have the following Lie algebra which is generated by $\{L_n|n\geq 0\}.$ It satisfies the following commutation rule $$\Big[ L_i ,L_j \Big]= (i-j)(L_{i+j}-\frac14 L_{i+j-1}).....*$$ My question is ...
40 views

### Why do the matrix elements of an operator correspond to the Fourier components of the observable in Heisenberg's Matrix Mechanics?

It is well-known that Heisenberg $a$ began developing his Matrix Mechanics by creating matrix components $$A(n,n-a,t)=A(n,n-a)e^{i\omega(n,n-a)t}$$ or $$A_{nm}(t)=A_{nm}e^{\frac{i}{\hbar}\omega(nm)t}$$...
2k views

453 views

### What does it mean for a unit vector to have a magnitude of 1?

Imagine a Cartesian coordinate system whose origin is associated with two unit vectors, ê and â, in a 2D-space. Now, let 0.5 cm be the unit of length in this coordinate system. The magnitude of a ...
87 views

### Unit Vectors in physics

I'm reading the Massachusetts Institute of Technology: "Review of Vectors" , and I've found this: I can't see any relationship between the text that is highlighted in yellow and what's depicted in ...
119 views

### Rank of a density matix

I was just trying to understand the meaning of rank of a density matrix. I came across the following post, which says that the rank of density matrix is the number of non-zero eigenvalues. And for a ...
67 views

### Is the basis of the occupation numbers of bosonic system orthonormal?

I am interested in the calculation of a correlation function in the Fock space of a system of $N$ bosons. As a trace it would be convenient for me to sum over the elements of the occupation numbers ...
201 views

### How does the partial transpose operation look like in matrix form?

The link here gives a nice description of how partial trace looks in matrix notation. I want a similar explanation for the matrix partial-transposition. How does matrix partial-transposition operation ...
221 views

### Why the basis of vectors and one-forms can not be related through the metric as a vector and one-forms?

I know that basis vector and basis of one-forms are related through $$\tilde{e}^\mu \cdot \vec{e}_\nu = \delta^\mu _\nu .\tag{1}$$ However, the metric has the property that allows to convert ...
173 views

### Gram-Schmidt process and degenerate subspace of the solutions to the Schrodinger's equation

So I know that in QM each linear combination of a degenerate set of wavefunctions is also a solution to the Schrodinger's equation (SE). The degenerate wavefunctions must be orthogonal to the non-...
51 views

### On the dimensionality of the Hilbert space for a central potential [duplicate]

So, we have that for particles in a central potential, the wavefunction can be expressed as: $$\psi(\mathbf{x})=R(r)Y^m_l(\theta,\phi)$$ Explanation (not crucial in the question): My question is: ...
So I have got the following question: Show that the scalar product of two cartesian vectors $p_i\cdot q_i$ is invariant under coordinate transformations (orthogonal transformations) Now I know ...