All Questions
18 questions
4
votes
2
answers
242
views
Is the momentum wave function's square amplitude always time-invariant for a free particle? [closed]
I have noticed whenever working with free particles that the square amplitude of the momentum wave function $|\Phi(p)|^2$ ends up being time invariant, so I followed this chain of logic supporting the ...
- 85
11
votes
6
answers
2k
views
Why are expectation values of an observable important in QM?
I've been reading that expectation values of an observable is all what we can get and are the key quantities of the theory, but performing the same experiment many times would generate a distribution ...
- 183
9
votes
3
answers
1k
views
Dirac's definition of probability in quantum mechanics
I'm currently reading "The principles of quantum mechanics" by Dirac, and I'm having some trouble understanding some of his assumptions, because in the quantum mechanics course I'm following ...
- 435
1
vote
2
answers
116
views
How does the state operator relate to the complete set of basis vectors?
I quantum mechanics we can represent the state of a system $\vert\psi\rangle$ in some Hilbert space as a complete set of basis vectors $\vert n\rangle$;
\begin{equation}
\vert\psi\rangle=\sum_n^Nc_n\...
- 471
1
vote
1
answer
313
views
How are probabilities different from expectation value for projective measurements
For projective operators,
Suppose we have set of operators $\{P_m\}$ then according to measurement postulate, probability of getting result $m$ is (I believe $m's$ here are eigenvalue. Please correct ...
4
votes
3
answers
2k
views
Multiplication of probability in quantum mechanics
Consider a ket-space spanned by the eigenkets of an observable $A$ and let $B$ be an additional observable on the same ket-space. We can build a filter that only lets an eigenvalue $a$ of $A$ through ...
- 324
0
votes
1
answer
186
views
Interpretation of Born Rule In QFT
Can we born rule be used to find probability of a particle to exist in a region in QFT using the formula $\int_a^b \psi(x)\psi^*(x)dx$,where $\psi(x)$ is a fermionic field?
If yes, please provide ...
- 73
1
vote
0
answers
75
views
Can this shape of matrix elements in the path integral formalism be linked to some sort of expectation value?
This question is about expressions of the form
$$
\langle x_f, t_i | \hat{x}(t) | x_i, t_i \rangle = \frac{1}{N} \int_{x(t_i) = x_i}^{x(t_f) = x_f} \mathcal{D} x~x(t)e^{i S[x]}.
$$
In the following ...
- 6,980
1
vote
1
answer
123
views
What is the distribution for a function of different quantum observables?
Suppose we have a quantum mechanical particle prepared in a pure state $\psi$, and an apparatus that can measure the orbital angular momentum of the particle along a specified orthogonal axis ($x$, $y$...
- 164
4
votes
3
answers
3k
views
Probability distributions in quantum mechanics
In general, how does one calculate a probability distribution, $P(A)$ for some observable $\hat{A}$, given a state $|{\psi}\rangle$?
I know that the expectation value of $A$ for $|\psi\rangle$ is ...
- 245
0
votes
1
answer
65
views
Effect of commuting observables on the probability of measuring a certain value [closed]
Say you can measure $3$ observables $(A, B, C)$ and you do the measurements in two different ways.
$\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\bra}[1]{\langle#1|}
\newcommand{\braket}[2]{\langle#1|#...
0
votes
1
answer
113
views
Generalisation of the measurement postulate in quantum mechanics
Given an observable that has a partially discrete and partially continuous spectrum of eigenvalues associated to it with the order of the spectrum's degeneracy being greater than 1, how would you ...
- 1,001
1
vote
1
answer
237
views
Analogy expectation of an observable / random variable
I'm trying to figure out the analogies between the expectation of a random variable $X$ and the expectation of an observable of a quantum mechanical system $A$ (using this wikipedia article).
The ...
- 334
4
votes
3
answers
2k
views
Positional probability density for combined spin and position states
In one dimension, given a particle in a quantum state $| \psi\rangle$, the probability density of position is given as $| \psi(x) |^2 = \psi^*(x) \psi(x) =\langle x | \psi \rangle\langle \psi | x \...
- 1,053
23
votes
7
answers
3k
views
Why is a Hermitian operator a "quantum random variable"?
To me, as a stupid mathematician, a random variable is a measurable function from some probability space $(\Omega, \sigma, \mu)$ to $(\Bbb{R}, B(\Bbb{R}))$. This makes sense. You have outcomes, events,...
user79317
4
votes
2
answers
503
views
If a quantum state is pure why are its observables still probabilistic?
As I understand it, a pure quantum state is one that can be represented as a ket $\lvert\psi\rangle$ in a Hilbert space, and it contains all the information about the state of the system. As such, we ...
- 3,267
2
votes
3
answers
911
views
Is the probability current an observable?
Is the probability current in Quantum Mechanics an observable? If so, how can it me measured (directly or indirectly)?
- 7,860
13
votes
2
answers
10k
views
Particle in a 1-D box and the correspondence principle
Consider the particle in a 1-d box, we know very well the solutions of it. I'd like to see how the correspondence principle will work out in this case, if we consider position probability density ...
- 2,172