Linked Questions

0
votes
1answer
117 views

Analysis of Harmonic Oscillator [duplicate]

In griffiths, Why was the lowest chosen like that while in the ladder analogy, the lower energies are like $a_-^n\psi$? That is, why didnt he chose the lowest rung to be $$a_-^n\psi$$
17
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3answers
6k views

Origin of Ladder Operator methods

Ladder operators are found in various contexts (such as calculating the spectra of the harmonic oscillator and angular momentum) in almost all introductory Quantum Mechanics textbooks. And every book ...
20
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5answers
2k views

What does periodicity of $e^{-iHt/\hbar}$ mean in physical terms?

The unitary time evolution operator $U(t)=e^{-iHt/\hbar}$ has some distinct flavour of periodicity to it because of $e^{x+2\pi i}=e^x$. Is this periodicity reflected in any way in physical systems? ...
17
votes
2answers
6k views

Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?

The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
17
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6answers
1k views

Is the harmonic oscillator potential unique in having equally spaced discrete energy levels?

I was wondering if the good old quadratic potential was the only potential with equally spaced eigenvalues. Obviously you can construct others, such as a potential that is infinite in some places and ...
6
votes
2answers
2k views

Proving that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space

How can I prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?
7
votes
1answer
4k views

Representation of operators in quantum mechanics

For which systems we represent the Hamiltonian as a differential operator and for which system we represent it by a matrix? Can the momentum be represented by a matrix operator?
6
votes
1answer
2k views

What physical significance has the Heisenberg Group?

I read that the canonical commutation relation between momentum and position can be seen as the Lie Algebra of the Heisenberg group. While I get why the commutation relations of momentum and momentum, ...
6
votes
2answers
581 views

An alternative definition of the creation and annihilation operators?

Suppose we have a system of bosons represented by their occupation numbers $$\tag{1} | n_1, n_2, ..., n_\alpha, ... \rangle$$ Then we can define creation and annihilation operators $$\tag{2} a_\alpha^\...
2
votes
1answer
305 views

Different hamiltonians for quantum harmonic oscillator?

The Hamiltonian for a classical simple harmonic oscillator is $$ H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$ With the usual choice of the ladder operators $$a = \frac{1}{\sqrt{2m\omega\hbar}}(m\...
0
votes
2answers
67 views

Proportionality of states in quantum harmonic oscillator

What is the justification for $a_{\pm} \psi_{n}$ being proportional to $\psi_{n\pm1}$ in a quantum harmonic oscillator? Here $a_{\pm}$ is the raising/lowering ladder operator.
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0answers
52 views

Eigenvectors and Eigenvalues of $a^\dagger a$ [duplicate]

A common question I have seen on problems sets and on exam papers is as follows (own words): Show that $a^\dagger$ and $a$ satisfy the canonical commutation relations. Hence show write down the ...