Linked Questions

2
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1answer
1k views

Why isn't the time-derivative considered an operator in quantum mechanics? [duplicate]

Based on my understanding when doing quantum mechanics we deal with a small set of mathematical objects: namely scalars, kets, bras, and operators. But then in the Schrodinger equation we have this ...
3
votes
1answer
567 views

Is there any Hamiltonian that contains time derivative? [duplicate]

Quantum mechanics is governed by Schrodinger's equation: $$\hat{H}\psi=i\hbar\partial_t \psi$$ It seems that Hamiltonian acts on wave functions like a time derivative. Just out of curiosity, is ...
1
vote
0answers
276 views

Hamiltonian in commutator contradiction [duplicate]

Consider the following: $$[ \hat H, \hat x]=\left[-\frac{\hbar^2 \hat p^2}{2m}+V,\hat x\right]\ne0 \text{ in general}$$ But $$[ \hat H, \hat x]=\left[i\hbar \frac{\partial }{\partial t},\hat x\right]...
-1
votes
1answer
86 views

Different forms for the energy operator [duplicate]

I am a little bit confused about which is really the energy operator. During the lectures, the professor told us that the energy operator is simply the Hamiltonian $\hat{H}$ and that the eigenvalues ...
0
votes
0answers
107 views

Definition of Hamiltonian in Quantum Mechanics [duplicate]

Is there any particular reason that the Hamiltonian operator was defined in quantum mechanics to be $$\hat H := \frac{\hat p^2}{2m} + V$$ as opposed to $$\hat H := i\hbar \frac{\partial}{\partial t}?$$...
0
votes
0answers
45 views

Non-hermitian hamiltonians and symmetry properties of $\partial_t$ [duplicate]

From this question another one came to my mind. Consider the Hilbert space $\mathcal{H}$ of function square integrables $\psi:\mathbb{R}^n\rightarrow\mathbb{C}$ with the usual inner product: $$ \...
86
votes
8answers
27k views

What is $\Delta t$ in the time-energy uncertainty principle?

In non-relativistic QM, the $\Delta E$ in the time-energy uncertainty principle is the limiting standard deviation of the set of energy measurements of $n$ identically prepared systems as $n$ goes to ...
27
votes
3answers
7k views

Time as a Hermitian operator in quantum mechanics

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 \...
27
votes
3answers
4k views

Is there an actual proof for the energy-time Uncertainty Principle?

As I understand, the energy-time uncertainty principle can't be derived from the generalized uncertainty relation. This is because time is a dynamical variable and not an observable in the same sense ...
8
votes
8answers
19k views

How to derive Schrödinger equation? [duplicate]

How is the Schrödinger equation $$ i\hbar\frac {\partial }{\partial t}\psi=H{\psi }$$ derived?
15
votes
4answers
2k views

Really how can an observable quantity be equal to an operator?

A wave-function can be written as $$\Psi = Ae^{-i(Et - px)/\hbar}$$ where $E$ & $p$ are the energy & momentum of the particle. Now, differentiating $\Psi$ w.r.t. $x$ and $t$ respectively, we ...
14
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4answers
3k views

Energy is actually the momentum in the direction of time?

By comparatively examining the operators a student concludes that `Energy is actually the momentum in the direction of time.' Is this student right? Could he be wrong?
10
votes
4answers
11k views

How is the hamiltonian a hermitian operator?

My book about quantum mechanics states that the hamiltonian, defined as $$H=i\hbar\frac{\partial}{\partial t}$$ is a hermitian operator. But i don't really see how I have to interpret this. First of ...
25
votes
2answers
3k views

Is there a time operator in quantum mechanics?

The question in the title has been asked many times on this site before, of course. Here's what I found: Time as a Hermitian operator in QM? in 2011. Answer states time is a parameter. Is there an ...
6
votes
3answers
899 views

An operator on the other side of the Schrödinger equation

A form of the Schrödinger equation is $$ \left[-\frac{\hbar^2}{2m} \nabla^2 + V(\vec{r}, t)\right]\Psi = i\hbar \frac{\partial}{\partial t} \Psi $$ The bracketed term is of course the ...

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