1
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2answers
65 views

Eigenvalues being physical observables

I think I'm comfortable with the PDE solutions to the Schrodinger equation. But as soon as we start putting these values in a matrix (in dirac notation), I lose my understanding and everything ...
1
vote
2answers
99 views

Commuting operators and Direct product spaces

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? When can an eigenket $|\lambda$1$\lambda$2$>$ ...
1
vote
2answers
47 views

How does Dirac conclude that $X_r(c_r)$ cannot vanish?

On page 32 of Dirac's book Principles of Quantum Mechanics, he considers the case when the linear, Hermitian$^1$ operator $\xi$ satisfies an algebraic equation $$\phi(\xi)\equiv(\xi - c_1)(\xi - ...
2
votes
1answer
72 views

Making an Incomplete Set of Observables Complete

In quantum mechanics, it seems a standard procedure that if you have an incomplete set of observables, then one can make this set complete by adding more commuting observables until the set becomes ...
0
votes
2answers
73 views

Differentiation operator with respect to observable acting as a function of the observable?

In his Principles of Quantum Mechanics Dirac writes: $$\int \langle \phi \frac{d}{dq}|q'\rangle dq' \psi(q')=\int \phi(q') dq' \frac{d\psi(q')}{dq'}.$$ To me it is rather strange, and it seems as if ...
0
votes
1answer
51 views

Observables in Quantum Mechanics

Studying on own quantum mechanics I came across: Preceeding text: A basic postulate of quantum mechanics tells us how to set up the operator corresponding to a given observable. Observables, ...
0
votes
2answers
69 views

Eigenstates of an observable

Can we use eigenstates of ANY observable as base of the Hilbert space? If we can, is this equal to the statement that those eigenstates are orthogonal to each other and normalizable?
0
votes
0answers
32 views

Experimental proof of the principle of superposition in QM [duplicate]

I have read that we need all operators in QM to be linear to confirm the principle of superposition which is experimentally well proven. I wonder how such an experiment could be made?
0
votes
2answers
67 views

How can $J_1^2, J_2^2, J_{1z}, J_{2z}$ commute mutually?

I'm reading through J. J. Sakurai's Modern Quantum Mechanics book and currently looking at the "Angular-momentum addition" part. Here, it says you have two options and that one option is to ...
2
votes
1answer
60 views

Probability of measuring two observables in a mixed state

Lets say i have density Matrix on the usual base $$ \rho = \left( \begin{array}{cccc} \frac{3}{14} & \frac{3}{14} & 0 & 0 \\ \frac{3}{14} & \frac{3}{14} & 0 & 0 \\ 0 & ...
1
vote
1answer
70 views

Is there any non-hermitian operator on Hilbert Space with all real eigenvalues?

The property of hermitian is the sufficient condition for eigenvalue being real. Is there any non-hermitian operator on Hilbert Space with all real eigenvalues? If there exist, then can all ...
6
votes
1answer
139 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.
0
votes
2answers
126 views

Quantum Mechanics - Observable

If $O$ represents an operator corresponding to an observable why does the following equality hold? $$\langle f(x)\, |\, O g(x)\rangle = \langle g(x) \,|\, O f(x) \rangle$$ It is used on the last ...
2
votes
1answer
166 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 ...
2
votes
1answer
232 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 ...
1
vote
1answer
172 views

Is kinetic energy in QM a state-property or is it distributed?

Suppose we have a quantum mechanical system, which is well described by its wave function in r-representation $\Psi$. We are interested in the properties of an observable, say the kinetic energy $T$. ...
2
votes
2answers
288 views

Do eigenvectors of quantum operators span the whole Hilbert Space?

I am trying to solve an exercise in Shankar's QM book (concretely 4.2.1), and I am asked the probability of each possible value for the operator $L_x$ when the particle is in a certain eigenstate of ...
2
votes
1answer
145 views

Why are Hermitian operators linked to observables?

In Quantum Mechanics, why is it that a self-adjoint operator is linked to an observable? What makes it measureable? And why isn't a non-Hermitian operator linked to an observable? Also, what type of ...
1
vote
0answers
62 views

How does linearity of a measurement imply that the commutator of all measured observables are $c$-numbers?

I really don't understand with the linearity conditions I have where this comes from.
1
vote
0answers
94 views

How functions become operators in quantum mechanics? [duplicate]

What used to be functions in the context of classical mechanics like position, linear momentum, angular momentum, etc in quantum mechanics are operators (these operators act on the state to get ...
2
votes
1answer
182 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 ...
3
votes
2answers
115 views

Observable Operator on a Superposition?

I'm probably missing something obvious and basic here but I can't make sense of certain usages of Observables as present in basic treatments of Quantum Mechanics that i've come across. $$ ...
0
votes
2answers
2k views

Expectation Values in Quantum Mechanics

Why is the expectation value what it is? Why don't you apply the operator, then multiply that by it's conjugate?
0
votes
2answers
2k views

What is the Momentum Operator?

I know the equation for the momentum operator, but what exactly is the momentum operator? It's bizarre to me that taking the derivative of the wave function, which is an operator, should return ...
2
votes
1answer
97 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 ...
8
votes
1answer
818 views

“An operator is hermitian”. Implications?

Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as: Every dynamical variable may be ...
15
votes
1answer
731 views

Intuitive meaning of Hilbert Space formalism

I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points: The observables are given by self-adjoint operators on the ...
6
votes
2answers
609 views

Eigenvalues of a quantum field?

Fields in classical mechanics are observables. For example, I can measure the value of the electric field at some (x,t). In quantum field theory, the classical field is promoted to an operator-valued ...
5
votes
3answers
4k views

Proof of Canonical Commutation Relation (CCR)

I am not sure how $QP-PQ =i\hbar$ where $P$ represent momentum and $Q$ represent position. $Q$ and $P$ are matrices. The question would be, how can $Q$ and $P$ be formulated as a matrix? Also, what is ...
3
votes
2answers
273 views

Why does $i ( LK-KL )$ represent a real quantity?

According to my textbook, it says that $i( LK-KL )$ represents a real quantity when $K$ and $L$ represent a real quantity. $K$ and $L$ are matrices. It says that this is because of basic rules. ...
3
votes
2answers
248 views

How to express continuous values as a matrix

Usually a quantity of a matrix is defined as the eigenvalues of the matrix. If so, how can anyone express continuous values, as in Schrodinger picture, into a matrix?
4
votes
1answer
208 views

The difference between projection operators and field operators in QFT?

Is there a good reference for the distinction between projection operators in QFT, with an eigenvalue spectrum of $\{1,0\}$, representing yes/no measurements, the prototype of which is the Vacuum ...
5
votes
3answers
381 views

Some questions on observables in QM

1-In QM every observable is described mathematically by a linear Hermitian operator. Does that mean every Hermitian linear operator can represent an observable? 2-What are the criteria to say whether ...
4
votes
1answer
2k views

Compatible Observables

My QM book says that when two observables are compatible, then the order in which we carry out measurements is irrelevant. When you carry out a measurement corresponding to an operator $A$, the ...
23
votes
5answers
2k views

Is mass an observable in Quantum Mechanics?

One of the postulates of QM mechanics is that any observable is described mathematically by a hermitian linear operator. I suppose that an observable means a quantity that can be measured. The mass ...
10
votes
5answers
12k views

What is the Physical Meaning of Commutation of Two Operators?

I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable ...
14
votes
2answers
2k views

Time as a Hermitian operator in QM?

In non-relativistic QM, on one hand we have the following relations: $$\langle x | P | \psi \rangle ~=~ -i \hbar \frac{\partial}{\partial x} \psi(x),$$ $$\langle p | X | \psi \rangle ~=~ i \hbar ...