Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy and the uncertainty principle and is generally used in single body systems. Use the ...
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36 views
Unitary invariance [migrated]
Why is it that for any non-negative matrix $M$ and unitary matrix $U$, we have
$$\sqrt{UMU^\dagger}=U\sqrt{M}U^\dagger$$?
This question has to do with Problem 2c from this sheet. I think I am ...
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0answers
30 views
Quantum fluctuation [on hold]
Is there any simple explanation for Quantum fluctuation? how particles can appear from vacuum? and how it may have been very important in the origin of the structure of the universe?
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23 views
What is a Zero-Phonon Line (ZPL)?
I am trying to the electronic structure of the negatively charged NV centre in diamond, where there is a so-called Zero-Phonon Line (ZPL). Can anybody explain what a ZPL is?
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2answers
51 views
Area under the graph of squared wave function
I was given a graph of square of the wave function of a hydrogen atom, against the distance of the electron from the nucleus (denoted by r).
What I know is that the square of the wave function gives ...
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1answer
64 views
How to derive or justify the expressions of momentum operator and energy operator?
It has been noted here$\! { \, }^{\text(1, 2)}$, for instance, that
$$\mathbf{F} = \frac{d}{dt}\!\!\biggl[ \, \mathbf{p} \, \biggr]$$
is true in all contexts.
Likewise,
in notable contexts it is ...
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1answer
30 views
Harmonic Oscillator Expectation Value
In Calculating the expectation value of the quantum harmonic oscillator, I've come across a problem for finding $\left \langle x \right \rangle$ for the coherent state $\left| \alpha \right \rangle$
...
2
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0answers
17 views
Is there a formalism for talking about diagonality/commutativity of operators with respect to an overcomplete basis?
Consider a density matrix of a free particle in non-relativistic quantum mechanics. Nice, quasi-classical particles will be well-approximated by a wavepacket or a mixture of wavepackets. The ...
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1answer
25 views
Selection rules in Stark effect
The energy level of an electron could be shifted by an electric field. $\langle n, l,m|[L_z,z]|n^{\prime},l^{\prime},m^{\prime}\rangle=(m-m^{\prime})\hbar \langle n, ...
4
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3answers
106 views
Why is ground state $| 0 \rangle$ of harmonic oscillator a coherent state?
Is ground state $| 0 \rangle$ of harmonic oscillator a coherent state just because it minimize the uncertainty product?
What is the intuition of this. I don't quite understand the significance of the ...
0
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1answer
44 views
Why does $\langle a_-\alpha|\alpha\rangle = \alpha $for harmonic oscillator
Why does $\langle a_-\alpha|\alpha\rangle = \alpha$ doesn't the ladder operator lower the $\alpha$ so that it became $\sqrt{\alpha}*\delta_{\alpha, \alpha_{-1}}$
Maybe its because that \alpha can be ...
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1answer
38 views
Time-ordered calculation for equal time
I have a question about how to calculate the following expectation value:
$$\langle0|\mathcal{T}\{{a^{\dagger}}(0,0) a(0,0)\}|0\rangle$$
where $|0\rangle$ is the ground state and $a^{\dagger}(x,t)$ ...
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2answers
77 views
The harmonic oscillator - ladder operators
Reading from Griffiths. I have got two questions.
First, the halmiltonian operator that used to find the energy eigenvalue in only harmonic oscillator is:
$$H={\hbar}w(a_-a_+-\frac{1}{2})$$ and ...
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1answer
45 views
Perturbations in arbitrary dimensions
In general is it acceptable to say that if a perturbation is in only one spatial direction then the energy eigenvalue to second order is only changed in that spatial direction?
For example 3D ...
0
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1answer
31 views
Harmonic Oscillator Energy to Momentum Expectation Value
If we are given a wave function written in terms of harmonic oscillator energy eigenfunctions how can we determine the maximum possible momentum expectation value? It's a combination of the first two ...
4
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1answer
69 views
Does the Time Evolution Operator Commute with any Other Operators?
Does the time evolution operator in quantum mechanics commute with any other operators, with a commutator of zero? Also, what exactly is the utility of the time evolution operator, is it more ...
