# Tag Info

5

A superposition of stationary states is not a stationary state. Suppose we have two kets, $| E_1 \rangle$ and $|E_2 \rangle$, which solve the TISE like so: $$\hat{H} | E_i \rangle = E_i | E_i \rangle$$ Here, the left hand side is the Hamiltonian operator and and the right hand side just shows that the stationary state picks up an eigenvalue. This is the ...

4

Consider two charges $q_1$ and $q_2$ kept at some separation. Suppose we want to calculate the potential energy of the system. By definition, potential energy is the work done to assemble such distribution. We can assemble the system in two ways: Bring $q_1$ to its place; no work done during this as there is no field present. Then bring $q_2$ to its ...

3

It's the time independent Schrödinger equation written using Dirac notation.

3

I'm not sure how you got that objective into your head: almost certainly that is not what you introductory second quantization text is telling you to do. Wikipedia has decent summaries of the bridge between QFT and QM, namely real scalar field theory ; multidimensional quantum oscillator; lattice phonon oscillators. The 1+1 quantum field $\psi$ you wrote is ...

2

The field and the wavefunction look similar, but they don't really have much to do with each other. The main point of the field is to group the creation and annihilation operators in a convenient way, which we can use to construct observables. As usual I will start with the free theory. If we want to find a connection to non-relativistic QM, the field ...

2

We don't control the allowed energies $E_i$ independently of the potential: the energies must be the eigenvalues of the Hamiltonian. The "inputs" are the shape and height of the barrier between the two wells. You can kinda sorta think of the energy difference between the symmetric state (with energy $E_1$ in your diagram) and the antisymmetric state (with ...

2

A double well with a high or wide barrier will have a smaller $\Delta E=E_2−E_1$ than one with a low or narrow barrier. (Less coupling.) I think we can understand this intuitively as follows but first it has to be said that rob is right: the energies $E_i$ are NOT inputs but the eigenvalues of the Schrödinger equation. Width, height and potential of the ...

2

The point is that the equation of motion of the fields is different if referring to temporal derivatives. In relativistic field theory, it is a second-order one and you need two initial conditions i.e. $\pi$ and $\pi$ to solve it. Quantizing, and interpreting the Fourier coefficients of the initial conditions as creation and annihilation operators, ...

2

I'm as confused as you by the boxed equation. At best the author is making that all-too-common mistake of reordering the expressions in a transitive equals relation, making the equation nonsensical when read left to right. However, it is not quite a tautology to prove what I think this is trying to prove: If $E$ and $\psi$ satisfy $H \psi = E \psi$, ...

2

It is a mathematical theorem that self-adjoint operators in Hilbert space have a complete spectrum. Note that "self-adjoint" has a special mathematical meaning. Not every Hermitian symmetric operator is self-adjoint. For example, the 1D free Schrodinger Hamiltonian on an open interval without boundary conditions is not self-adjoint. The reason is that we ...

2

The "time-independent Schrödinger equation" is just an equation for the eigenvalues and eigenvectors of the Hamiltonian operator on the Hilbert space of states (typically $L^2(\mathbb{R}^3,\mathrm{d}x)$, the "space of wavefunctions") The spectral theorem tells us that the eigenvectors of any self-adjoint operator form a basis for the space the operator ...

2

The matrix element $\langle x |\psi\rangle$ where $x$ is allowed to vary is just the wavefunction $$\psi(x) = \langle x |\psi\rangle.$$ So the matrix element $\langle x |p|\psi\rangle$ is just the wavefunction representation of the state $p|\psi\rangle$. We know that the momentum operator in the wavefunction representation acts as a derivative, so ...

1

You seem to be thinking that the electron has a Coulomb potential energy and the nucleus has a separate Coulomb potential energy. That's not how you should think. Potential energy belongs to the whole system interacting with other parts of the system. Here we have an electron interacting with a proton. That system has potential energy. If we were to add ...

1

Your Transformation is wrong and it is easy to see that. Your transformed derivative of $\partial_{r_2}$ does not depend on the $r = r_1-r_2$ part of the transformation at all. In other words with your logic I could equally write $\partial_{r_2} = \frac{\partial r}{\partial r_2} \partial_{r} = - \partial_{r}$. The correct transformation is \partial_{r_2} = ...

1

I would like to begin by your last question. I am of the opinion that this is correct. I am not versed in systems engineering but I see the equivalence in the mathematical treatments. And QM is very much about the mathematics, is more a descriptive body of knowledge than it is an explanatory. And proof of this is the many interpretations it has, all of ...

1

Comments to the question (v3): Eqs. (1) are part of the CCR for a scalar field, such as, e.g., a real or complex Klein-Gordon field, a Schrödinger field, etc. Eq. (2) refers to the Schrödinger field, which is a complex field, see e.g. this Phys.SE post. A real Schroedinger field does naively not make sense since e.g. the expected kinetic term $\propto ... 1 If I am correctly interpreting your question, you have a 1D time independent S. equation with$V$which is discontinuous in some points. You solve that equation for a fixed eigenvalue$E$separately in each continuity interval obtaining functions$\psi_E$which are$C^2$in every open interval and depends on two arbitrary constants. Finally you mach the ... 1 The Schrodinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but given ... 1 The problem is considering an incoming right-mover for$x<0$and asks how it scatters off a step potential into a reflected outgoing left-mover for$x<0$and a transmitted outgoing right-mover for$x>0$. The last possibility -- an incoming left-mover for$x>0\$ -- is not present in this scattering experiment. That's the answer to OP's question. ...

1

Just so this doesn't slip past: Now I think if one side of the potential barrier is higher, the particle will be more likely to spend time on the left side than the right side, so the wavefunction should have higher amplitudes on the left (skewed to the left): This is incorrect. Between A and B the well is deeper, so the particle goes faster. Between B ...

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