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# Questions tagged [linear-algebra]

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

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### What is the trace of the (spatial) projection tensor?

In the discussion on gravitational waves in Moore's A General Relativity Workbook, Box 33.4 on page 391 introduces the projection operator $$P^j_m\equiv\delta^j_m - n^jn_m$$ where $\vec n$ is a unit ...
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### Spin 1/2 system in the algebraic approach to quantum mechanics

I'm trying to understand the $\ast$-algebra approach to QM and QFT, and so I have decided to first try to understand how this works in one of the simplest systems: a particle with spin 1/2. This is ...
128 views

### Why does angular momentum point up for a counterclockwise rotation? Why not down?

I am a high school student, and lately, I am founding things a bit perplexing on some topics concerning cross product in physics. In angular momentum, we learned that the direction of an object ...
283 views

### Ladder operator identity for $\langle n | (a+a^\dagger)^k | m \rangle$

I would like to know if there is a convenient identity (and what it is) for $$\langle n | (a+a^\dagger)^k | m \rangle$$ where $| n \rangle, \, | m \rangle$ are energy eigenstates of a simple ...
140 views

### Postulates of inner product

In quantum mechanics, two fundamental properties of inner products (J.J Sakurai) Chapter 1.2, are: $\langle \alpha|\beta\rangle = \langle \beta|\alpha\rangle^*$ $\langle \alpha|\alpha\rangle \ge 0$ ...
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### What is norm of matrix element in Fermi Golden Rule

Fermi Golden Rule says: $\Gamma \propto |M_{ij}|^2$ I know how to get $M_{ij}$, but how do I proceed? How do I take a norm of Hermitian matrix? There is no clear (to me) definition in the internet ...
329 views

### Eigenvectors and eigendecomposition of Pauli matrices, why isn't there many?

Say we are finding eigenvectors of $\sigma _z$, the eigenvalues are $1,-1$ so filling into the eigenvalue equation $\sigma _z (a,b)=(a,-b)=1(a,b)$ and we find that $b=0$. I am confused about why we ...
215 views

### Constructing a CPTP-map on one density matrix using another

My question is: If one is given two density matrices $A$ and $B$, is there a way to use the first to construct a CPTP-map (quantum channel) acting on the on the other? I thought that Stinespring ...
313 views

### Why is the projection operator corresponding to $\tilde M$ given by $P_m\otimes I_B$?

Nielsen and Chuang, Chapter 2 (Box 2.6): Suppose $M$ is any observable on a system $A$, and we have some measuring device which is capable of realizing measurements of $M$. Let $\tilde M$ ...
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### Which set of basis states can a quantum system of qubits actually collapse to?

I was watching a video on "How Does a Quantum Computer Work?". I'm confused about what they mean by: "Although the qubits can exist in any combination of states, when they are measured they must ...
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### Uniqueness of simultaneous eigenstates of two linear operators

I was solving a homework problem where the question gives the representation of two operators in matrix form, in some arbitrary set of basis vectors. It then asks to find the simultaneous eigenstates ...
84 views

### Doubt in the transformation in R Shankar's Quantum Mechanics

On page number $19$ of the book Principles of Quantum Mechanics by R. Sankar, The author describes a transformation, essentially a rotation of the axes by an angle of $\frac{\pi}{2}$ counterclockwise, ...
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### Requirements for the inner product in Hilbert space

I was reading the following text about the mathematical foundations of quantum mechanics when I stumbled upon the following conditions that the inner product must satisfy: I woud like to understand ...
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### Velocity vector $= 2î +3j +4k$. How is this vector 1D despite all the three axes being involved?

velocity vector = 2î + 3j + 3k How can this vector be 1D despite there being all the three axes involved
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### Solve eigenvalue problem with known constraint on one of the Eigenvalues

I have the following problem and would appreciate any help. I have a real, symmetric matrix M given by M=\begin{pmatrix} m_{11} & m_{12} & m_{13} & m_{14} \\ m_{12} & m_{...
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### Does the dimensions of the wavefunction vector in Hilbert space depend on the number of eigenfunctions it is a superposition of?

I've seen people say that wavefunctions represented as vectors in a Hilbert space can (but don't have to) have infinite dimensions. So if a state vector requires X number of basis eigenfunctions ...