While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think...
I have always learnt that a Hermitian operator in non-relativistic QM can be treated as an "experimental apparatus" ie unitary transformation, measurement, etc.
However this makes less sense to me in QFT. A second-quantised EM field for instance, has field operators associated with each...
That's my attempting: first I've wrote ##e## in terms of the power series, but then I don't how to get further than this $$ \sum_{n=0}^\infty (-1)^n \frac {Â^n} {n!} \hat B \sum_{n=0}^\infty \frac {Â^n} {n!} = \sum_{n=0}^\infty (-1)^n \frac {Â^2n} {\left( n! \right) ^2} $$. I've alread tried to...
Hi!
When calculating ##(\hat{a} \hat{a}^{\dagger})^2## i get ##\hat{a} \hat{a} \hat{a}^{\dagger} \hat{a}^{\dagger}## which is perfectly fine.
But how do I end up with the ultimate simplified expression $$\hat{ a}^{\dagger} \hat{a} \hat{a}^{\dagger} \hat{a} + \hat{a}^{\dagger} \hat_{a} +...
1. Homework Statement
Consider two operators A and B, such that [A,[A, B]] = 0 and [B,[A, B]] = 0 . Show that
Exp(A+B) = Exp(A)Exp(B)Exp(-1/2 [A,B])
Hint: define Exp(As)Exp(Bs) as T(s), where s is a real parameter, differentiate T(s) with respect to s, and express the result in terms of...
1. Homework Statement
After proving the relations ##[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0## and ##[\hat{b}_i,\hat{b}_j]=0##, I want to prove that ##[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}##, however I'm not sure where to begin.
2. The attempt at a solution
I tried to apply the...
1. Homework Statement
Hi, guys. The question is: For a 3-state system, |0⟩, |1⟩ and |2⟩, write the matrix representation of the raising operators ## \hat A, \hat A^\dagger ##, ## \hat x ## and ##\hat p ##.
2. Homework Equations
I know how to use all the above operators projecting them on...
> Operator $$\hat{A}$$ has two normalized eigenstates $$\psi_1,\psi_2$$ with
> eigenvalues $$\alpha_1,\alpha_2$$. Operator $$\hat{B}$$ has also two
> normalized eigenstates $$\phi_1,\phi_2$$ with eigenvalues
> $$\beta_1,\beta_2$$. Eigenstates satisfy:
> $$\psi_1=(\phi_1+2\phi_2)/\sqrt{5}$$
>...
This may seem rather silly, but how would I go about enunciating Ehrenfest’s theorem?
Also, does anyone know what this theorem implies for the relation between classical and quantum mechanics?
Any suggestions or help is greatly appreciated!
Hi, why do unbounded operators and bounded operators differ so much in terms of defining their spectra?
1. The unbounded operator requires a self-adjoint extension to define its spectrum.
2. A bounded one does not require a self-adjoint extension to define the spectral properties.
3. Still the...
Hi, in a text provided by DrDu which I am still reading, it is given that "the momentum operator P is not self-adjoint even if its adjoint ##P^{\dagger}=-\hbar D## has the same formal expression, but it acts on a different space of functions."
Regarding the two main operators, X and D, each has...
I am interested in defining Krauss operators which allow you to define quantum measurements peaked at some basis state. To this end I am considering the Normal Distribution. Consider a finite set of basis states ##\{ |x \rangle\}_x## and a set of quantum measurement operators of the form $$A_C =...
Hi, I have in a previous thread discussed the case where:
\begin{equation}
TT' = T'T
\end{equation}
and someone, said that this was a case of non-linear operators. Evidently, they commute, so their commutator is zero and therefore they can be measured at the same time. What makes them however...
Hi, I noticed that the raising and lowering operators:
\begin{equation}
A =\frac{1}{\sqrt{2}}\big(y+\frac{d}{dy}\big)
\end{equation}
\begin{equation}
A^{\dagger}=\frac{1}{\sqrt{2}}\big(y-\frac{d}{dy}\big)
\end{equation}
can be used to solve the eqn HY = EY
However I am curious about...
The following is a somewhat mathematical question, but I am interested in using the idea to define a set of quantum measurement operators defined as described in the answer to this post.
Question:
The Poisson Distribution ##Pr(M|\lambda)## is given by $$Pr(M|\lambda) =...
Hi, I have an operator which does not obey the following condition for boundedness:
\begin{equation*}
||H\ x|| \leqslant c||x||\ \ \ \ \ \ \ \ c \in \mathscr{D}
\end{equation*}
where c is a real number in the Domain D of the operator H.
However, this operator is also not really unbounded...
In Hartree-Fock method, I saw the Fock operator has two integrals: Coulomb integral and exchange integral. One can define two operator. "The exchange operator is no local operator" why? Whats de diference: local and no local operator?
And why do the operators have singularities?
thanks
Given a wave function $$\Psi(r,\theta,\phi)=f(r)\sin^2(\theta)(2\cos^2(\phi)-1-2i*\sin(\phi)\cos(\phi))$$ we are trying to find what a measurement of angular momentum of a particle in such wave function would yield.
Attempts were made using the integral formula for the Expectation Value over a...
Suppose ##A## is a linear operator ##V\to V## and ##\mathbf{x} \in V##. We define a non-linear operator ##\langle A \rangle## as $$\langle A \rangle\mathbf{x} := <\mathbf{x}, A\mathbf{x}>\mathbf{x}$$
Can we say ## \langle A \rangle A = A\langle A \rangle ##? What about ## \langle A \rangle B =...
Hi at all, i've a curiosity about the role that the weight function w(t) she has, into the define of adjoint & s-adjoint op.
It is relevant in physical applications or not ?
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Hi at all
On my math methods book, i came across the following Fredholm integ eq with separable ker:
1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi
With integral ends(0,pi/2)
I do not know how to proceed, for the solution...
Hi All,
Perhaps I am missing something. Schrodinger equation is HPsi=EPsi, where H is hamiltonian = sum of kinetic energy operator and potential energy operator. Kinetic energy operator does not commute with potential energy operator, then how come they share the same wave function Psi???? The...
1. Homework Statement
Question:
(With the following definitions here: )
- Consider ##L_0|x>=0## to show that ##m^2=\frac{1}{\alpha'}##
- Consider ##L_1|x>=0 ## to conclude that ## 1+A-2B=0##
- where ##d## is the dimension of the space ##d=\eta^{uv}\eta_{uv}##
For the L1 operator I am able...
I am reading a proof of why
\left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z
Given a wavefunction \psi,
\hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...
In classical mechanics, the Hamiltonian and the Lagrangian are Legendre transforms of each other. By analogy, in quantum mechanics and quantum field theory, the relationship between the Hamiltonian and the Lagrangian seems to be preserved. Where can I find a derivation of the Lagrangian...
I want to clarify the relations between a few different sets of operators in a conformal field theory, namely primaries, descendants and operators that transform with an overall Jacobian factor under a conformal transformation. So let us consider the the following four sets of operators...
When I learnt about operators, I learnt <x> = ∫ Ψ* x Ψ dx, <p> = ∫ Ψ* (ħ/i ∂/∂x) Ψ dx. The book then told me the kinetic energy operator
T = p2/2m = -ħ2/2m (∂2/∂x2)
I am just think that why isn't it -ħ2/2m (∂/∂x)2
Put in other words, why isn't it the square of the derivative, but...
1. Homework Statement
Vectors I1> and I2> create the orthonormal basis. Operator O is:
O=a(I1><1I-I2><2I+iI1><2I-iI2><1I), where a is a real number.
Find the matrix representation of the operator in the basis I1>,I2>. Find eigenvalues and eigenvectors of the operator. Check if the eigenvectors...