In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

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Applying an operator to a function vs. a (ket) vector

I have a question regarding the effect of quantum mechanical operators. The definition that I'm familiar with says that an operator $A$ acts on a vector from a Hilbert space, $|\psi\rangle$, and the ...
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463 views

Self-adjoint and unbounded operators in QM

An operator $A$ is said to be self-adjoint if $(\chi,A\psi)=(A\chi,\psi)$ for $\psi, \chi \in D_A$ and $D_A=D_{A^\dagger}$. But for the free particle momentum operator $\hat{p}$ these inner products ...
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572 views

Regularisation of infinite-dimensional determinants

Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM? Edit: I failed to make myself clear. In finite ...
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606 views

Does the commutator of anything with itself not vanish?

In a quantum mechanics exam one question was to write the commutator of a couple of operators. Everybody got points taken away since they did not write $[Q_i, Q_i] = 0$ for all the operators $Q_i$ in ...
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427 views

The cleverest way to calculate $\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]$ with $\left[\hat{a},\hat{a}^{\dagger}\right]=1$

Who can provide me some elegant solution for $$\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]\qquad\text{with} \qquad\left[\hat{a},\hat{a}^{\dagger}\right]~=~1$$ other than brute force calculation? ...
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1answer
194 views

Directional derivatives in the multivariable Taylor expansion of the translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: ...
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1answer
240 views

Evaluate $\langle \mathbf{p} | 1/\hat{r} | \mathbf{p}' \rangle$

In Sakurai's Problem 1.27 b), we use $\langle \mathbf{r} | \mathbf{p}\rangle = e^{i\mathbf{p}\cdot\mathbf{r}/\hbar}$ to show that $$ \langle \mathbf{p} | F(\hat{r}) | \mathbf{p}' \rangle = ...
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Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?

I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
7
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351 views

Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?

Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$ where ...
7
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Proof for the completeness of eigenfunctions of a self-adjoint operator

I always heard the eigenfunctions of a self-adjoint operator form a complete basis. Where can I find a proof in infinite dimension space? Presumably readable for physicists.
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421 views

Non-associative operators in Physics

Are non-associative operators (or other kind of elements) used in Physics? For example, in QM I'm looking for something like this: $A(BC)|\psi\rangle \ne (AB)C|\psi\rangle$ NOTE: I think that this ...
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Implications of unbounded operators in quantum mechanics

Quantum mechanical observables of a system are represented by self - adjoint operators in a separable complex Hilbert space $\mathcal{H}$. Now I understand a lot of operators ...
3
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451 views

Commutator algebra in exponents

Considering $X$ and $Y$ such that $[X,Y]=\lambda$, which is complex, and $\mu$ is another complex number, prove: $$e^{\mu(X+Y)}=e^{\mu X} e^{\mu Y} e^{-\mu^2\lambda/2}$$ My attempt (so far) is: ...
2
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1answer
185 views

The Physical Meaning behind a Commutator [duplicate]

I've just been introduced to the idea of commutators and I'm aware that it's not a trivial thing if two operators $A$ and $B$ commute, i.e. if two Hermitian operators commute then the eigenvalues of ...
6
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1answer
167 views

Eigenstate of field operator in QFT

Why don't people discuss the eigenstate of the field operator? For example, the real scalar field the field operator is Hermitian, so its eigenstate is an observable quantity.
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143 views

Eigenstates of a Hermitian field operator

Consider a Hermitian field operator $\phi(x)$ with eigenstates satisfying $$ \phi(x) |\alpha\rangle = \alpha(x) | \alpha \rangle $$ I'm trying to determine the inner product between the eigenstates. ...
4
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1answer
184 views

Continuity domain for momentum operator

I know this is essentially a mathematic question, but I received no answer on math SE. Moreover it has a direct application in physics, so I thought to ask this here too. The momentum operator in one ...
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373 views

Unitary spacetime translation operator

Srednicki writes: We can make this a little fancier by defining the unitary spacetime translation operator $$ T(a) \equiv \exp(-iP^\mu a_\mu/ \hbar) $$ Then we have $$ T(a)^{-1} \phi(x) T(a) = ...
3
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1answer
125 views

Observables - what are they?

I often read in books that an observable is represented by an Hermitean operator. But it is deceiving as operator isn't the observable. As far as I've read the observable is denoted like $\langle ...
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1answer
464 views

State normalization in Dirac's formulation of quantum mechanics

Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $ Our ...
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What is the physical meaning of weak expectation values?

In the two-state formalism of Yakir Aharanov, the weak expectation value of an operator $A$ is $\frac{\langle \chi | A | \psi \rangle}{\langle \chi | \psi \rangle}$. This can have bizarre properties. ...
3
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2answers
122 views

Showing $K_\pm$ are raising/lowering operators

In this post, I have the following operators defined: $$K_1=\frac 14(p^2-q^2)$$ $$K_2=\frac 14 (pq+qp)$$ $$J_3 = \frac 14 (p^2+q^2)$$ I am given $ J_3|m\rangle = m|m\rangle$ and asked to show that ...
3
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2answers
230 views

Is there a four dimensional form of Born's Rule -redub

Generalizing Born's Rule for 4-dimensions $x_4$, write $$\langle a\rangle = \int\Psi A\Psi^* \mathrm{d}x_4$$ Is this consistent with quantum mechanics? Is this a generalized form of the Born's ...
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1answer
192 views

Help Simplifying a Commutator Equation

For the SHO, our teacher told us to scale $$p\rightarrow \sqrt{m\omega\hbar} ~p$$ $$x\rightarrow \sqrt{\frac{\hbar}{m\omega}}~x$$ And then define the following $$K_1=\frac 14 (p^2-q^2)$$ $$K_2=\frac ...
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1answer
188 views

What are the Time Operators in Quantum Mechanics? [duplicate]

I don't understand at all what the time operators are in quantum mechanics. I thought that given a wave function, because it's a function of time, we could simple put in any time in the future to find ...
2
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1answer
482 views

Quantum mechanical analogue of conjugate momentum

In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ...
2
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1answer
1k views

A Short way to show Conservation of Quantum Laplace–Runge–Lenz Vector?

I had been asked to prove the conservation of Quantum Laplace–Runge–Lenz Vector: ...
2
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2answers
179 views

Matrix representing the quantity - why can some matrices not be physical quantity?

In Heisenberg picture, my textbook says that the following matrix $A = \frac{5}{3}\Sigma_1 + i\frac{4}{3}\Sigma_2$ cannot represent physical quantity. the book says this is because ...
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5answers
794 views

Observable: possible outcome of measurement vs (linear) transformation

One of the postulates of quantum mechanics is that every physical observable corresponds to a Hermitian operator $H$, that the possible outcomes of the measurements are eigenvalues of the operator, ...
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5answers
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How to get the position operator in the momentum representation from knowing the momentum operator in the position representation?

I know that $$\tag{1}\hat{p}~=~-i\hbar \frac{\partial}{\partial x}~.$$ How can I get $$\tag{2}\hat{x}~=~i\hbar \frac{\partial}{\partial p}~?$$ I think this simple and I'm just over thinking it, ...
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151 views

What is the analogy of $|x\rangle$ in quantum field theory?

Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ ...
3
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4answers
549 views

Annihilation and Creation operators not hermitian

The annihilation and creation operators are given below $$ \hat a|n\rangle=\sqrt{n}|n-1\rangle\qquad\text{and}\qquad\hat a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle $$ Now I know the operator a^ is not ...
3
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429 views

Derivative of a Position Eigenket

I was flicking through Zettili's book on quantum mechanics and came across a 'derivation' of the momentum operator in the position representation on page 126. The author derived that ...
3
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1answer
65 views

Linearizing Quantum Operators [duplicate]

Possible Duplicate: Linearizing Quantum Operators I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator. $$ ...
2
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1answer
349 views

Non-Hermitian operator with real eigenvalues?

So we know that in Quantum Mechanics we require the operators to be Hermitian, so that their eigenvalues are real ($\in \mathbb{R}$) because they correspond to observables. What about a non-Hermitian ...
2
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2answers
2k views

How To Use Ladder Operators?

I'm studying for a test in quantum mechanics and I'm having a hard time understanding how to use ladder operators. There are no examples in my text book, only definitions that I can't understand how ...
2
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2answers
128 views

Identity of Operator Product Expansion (OPE)

I have one more s****d question in Polchinski's string theory book, Eqs. (2.3.14a) $$ j^{\mu}(z) :e^{ik \cdot X(0,0)}:~ \sim~ \frac{k^{\mu}}{2 z} :e^{ik \cdot X(0,0)}:,$$ where $j^{\mu}_a ...
2
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2answers
274 views

Physical meaning of some operators formed by $|Q\rangle \langle Q|$

In Dirac's formulation of quantum mechanics, Suppose that $q$ represents position observable. About $|q\rangle \langle q|$: what does this operator mean? I do get that it results in an operator, but ...
2
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1answer
70 views

A physical quantity that is a real combination and commutability

Suppose that a matrix $$A ~=~ x_1 B + x_2 C$$ is a linear combination of two self-adjoint matrices $B$ and $C$. I'm interested in when $A$ represents a physical quantity. When the linear ...
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1answer
68 views

How to write QM operator if I know all of it's eigenfunctions?

Suppose I have selected enough orthogonal functions in representation of operator A and I want to derive operator B which has ...
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3answers
95 views

Can one construct a new operator in terms of the powers of another operator?

Suppose we have a quantum state, well described by its time-independent wave function Psi. And we have a well-defined Hermitian (self-adjoint) operator $A$. We successfully evaluate the expectation ...
1
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1answer
377 views

Matrix representation for fermionic annihilation operator

My guess it should look something like this: $ c_\sigma = ...
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0answers
124 views

Time ordering and Fermions

Having time ordering operator for fermions, should it reverse sign if it swaps operators with opposite spin variable? In other words should $T[c_{t_1,\uparrow}c_{t_2,\downarrow}^\dagger]$ return ...
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0answers
55 views

Why there is no operator for time in QM? [duplicate]

Is there one central reason why there is no "Time" operator in QM? I know this question has been asked before, but I thought I would try to stimulate some fresh thinking.
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How is it shown that the composition of two real operators is generally not real? [duplicate]

Dirac on page 28 of his QM book writes: Thus the conjugate complex of the product of two linear operators equals the product of the conjugate complexes of the factors in the reverse order. As ...
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1answer
25 views

How to write “postselection” operator?

Suppose, I wish to know an operator, which eigenvalue is 1 if state is exactly F and 0 ...
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4answers
4k views

Product of exponential of operators

in the context of non-relativistic quantum mechanics I want to show that, for any $A$ and $B$ operators $$e^{A}e^{B}=e^{A+B} $$ if and only if $$[A,B]=0$$ I remember my professor told use about ...
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Antilinear operators [duplicate]

I'm a bit confused about antilinear operators: A (a|n>+b|m>)=(a* A|n>+b* A|m>) How follows now that ...