All Questions
Tagged with anticommutator quantum-mechanics
46
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(Anti) Commutation relation of derivative of the fermionic operator
While deriving Semiconductor Bloch equation, I stumbled upon a commutation relation that I have never seen before. It looks like,
$$[\alpha_k^{\dagger}\alpha_k, \alpha_{k'}^{\dagger}(\nabla_{k'}\...
- 1,007
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2
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Why does $[Q,P]=i\hbar$ work for fermion? Shouldn't fermion satisfy anticommuting relation?
For hydrogen, we use $[Q,P]=i\hbar$ for electron, which is a fermion. Does it have a deeper reason such as that we're really considering the proton + electron system, which might be of bosonic nature?
- 619
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Help with commutator algebra with fermionic operators
I am struggling to understand how the following is true for the fermionic creation/annihilation operators $a^\dagger, a$: $$[a^\dagger a, a]=-a$$
If someone could walk me through the math derivation ...
- 101
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Relationship between anti-commutators and correlation
Ballentine (in his solution at the back of the book to his Problem 8.10) writes that
$$[Tr(\rho \{A,B\}/2)]^2$$
is related to the correlation between the observables represented by $A,B$, but gives no ...
- 1,095
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Can Exceptional Jordan Quantum mechanics model field theory?
Exceptional Jordan Quantum mechanics is an interesting case in which observables are modelled with $3\times3$ Hermitian octonion matrices $\mathbb{J}_3(\mathbb{O})$. There is the Jordan product $A\...
user84158
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Commutation of kinetic energy operator with Hamiltonian
I am basically trying to calculate current energy operator $\hat{\mathbf{J}}_E(\mathbf{r})$ by using Heisenberg equation of motion as
$$
-\nabla\cdot \hat{\mathbf{J}}_E(\mathbf{r})=\frac{i}{\hbar}[H,\...
- 831
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Can Hadamard's formula be used for fermionic operators?
Can I use this special case of Hadamard's formula
$$e^\hat B \hat A e^{-\hat B}= A + [B,A]+\frac{1}{2!}[B, [B,A]] + \dots$$
for fermionic operators?
Suppose I have fermionic operators that obey ...
- 179
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Anticommutation and Bogoliubove transformation
I am given the following transformation:
\begin{equation}
\begin{bmatrix}
...
- 43
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What is the physical meaning of the anticommutator of two observables? [duplicate]
It is quite clear to me that when two operators commute it implies that two different observables associated with the respective operators can be measured simultaneously with the exact accuracy. But ...
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From Fermions to CAR
I do not understand why Fermions statistic is equivalent to the CAR. In other words, suppose we have $\{a_i\}_i$ set of Fermionic operators acting on an Hilbert space, they satisfy the Canonical ...
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Are there any 3 or more Hermitian solutions to the problem: $\alpha_i^2=1$, $\{\alpha_i, \alpha_j \}=2$
I’m trying to generate some matrices which are similar to Pauli’s but with the following anti-commutation relation
$$\{\alpha_i, \alpha_j\}=\alpha_i \alpha_j + \alpha_j \alpha_i = 2 \tag{1}$$
And
$$\...
- 2,241
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Commuting but not anti-commuting operators
Two Hermitian operators $\hat{A}$ and $\hat{B}$ are such that they commute but don't anti-commute. In this case, even they commute their uncertainty product will not be zero, is it right?
- 199
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Defining particles by their commutation/anti-commutation relations
In my studies of many-body physics, I have encountered three types of particles, which can be defined based on their commutation/anti-commutation relations.
Fermions, defined by raising/lowering ...
- 659
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Why is the anticommutator of the uncertainty principle omitted if it serves to increase the accuracy of our "knowledge" of a quantum state?
The generalized uncertainty principle can be derived and shown to be this which is fine and rigorous.
$\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle ...
- 207
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Creation and annihilation operators for fermions from anticommutator
In a question, I was given that $a^{\dagger}a + a a^{\dagger} =1$ and asked to show what $a|n\rangle$ and $a^{\dagger}|n\rangle$ would be, given that $H|n\rangle=(a^{\dagger}a + 1/2)$.
I am getting ...
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