Title: ETH_Sem3 / Physik_III / Woche9

Q: The vector space of quantum mechanical states:
A: Hilbert space

Q: Dirac notation:
Representation of a state vector
A: \(\ket{S(t)}\)

Q: Dirac notation:
Representation of a state in position basis
A: \(\psi(x,t) = \braket{x|S(t)}\)
bra: \(\bra x\)
ket: \(\ket{S(t)}\)

Dirac notation:
State in position representation (in different bases)
\(\braket{x|S(t)} =\) \(\int\psi(x,t)\delta(x-y)dx =\) \(\int \tilde{\psi}(p,t) \frac{e^{ipx/\hbar} }{\sqrt{2\pi \hbar} }dp =\) \(\sum_n C_n(t) \phi_n(x)\)
With expansion coefficients:
\(C_n(t) = \braket{e_n|S(t)} = \int_{-\infty}^\infty\phi_n^\star(x)\psi(x,t)dx\)

Q: Advantage of Dirac notation:
A: We no longer have to choose a basis to write our state

Q: QM equivalent to Matrix
A: Linalg equivalent to operators representing observables

Q: Matrix elements of an operator \(\hat{O}\) using \(\ket{\phi_N}\) as a complete basis
A: \(\braket{\phi_m|\hat{O}|\phi_n} \equiv O_{mn}\)

Q: [...]
In discrete basis
[...]
In continuous basis:
[...]
A: Orthogonality:; \(\braket{\phi_n|\phi_m} = \int \phi^\star_n(x)\phi_m(x)dx = \delta_{mn}\); \(\braket{\phi(\alpha)|\phi(\alpha\prime)} = \delta(\alpha-\alpha\prime)\)

Creating operators in dirac notation:
\(\ketbra{\alpha}{\beta}=\hat{O}\)
Identity for orthonormal basis:
\(\sum_n \ketbra{\phi_n}{\phi_n} = \hat{I} = 1\) (discrete)
\(\int\ketbra{\phi(\alpha)}{\phi(\alpha)}d\alpha = 1\) (continuous)

Q: Trick in dirac notation:
A: insert hidden \(1 = \int\ketbra{p}{p}\)

Eigenstates of an observable operator \(\hat{O}\) form a complete basis \(\{\ket{O_n}\} \) or \(\{\ket{O(\alpha)}\}\)
We label those eigenstates with their eigenvalues:
\(\hat{O}\ket{o_n} = o_n\ket{o_n}\) or \(\hat{O}\ket{o(\alpha)} = o(\alpha)\ket{o(\alpha)}\)
Any state \(\ket{\psi}\) can be written as:
\(\ket{\psi} = \sum_n C_n(t) \ket{o_n} \) or  \(\ket{\psi} = \int\psi(\alpha,t) \ket{o(\alpha)} d\alpha\)

Q: Probability of getting a certain result after measuring:
Discrete basis:
[...]
Continuous basis:
[...]
A: \(|\braket{o_n|\psi}|^2 = |C_n|^2\); \(|\braket{o(\alpha)|\phi}|^2 d\alpha = |\psi(\alpha,t)|^2d\alpha\)
Back Extra: The state of the system becomes the measurement. We call this collapse or projection.
In the continous case the "degree of collapse" depends on the measurement precision.

Definition: Commutator
\(\hat{O}\hat{Q} - \hat{Q}\hat{O} \equiv [\hat{O},\hat{Q}]\)
Two operators commute when their commutator is 0.
This means that they share a set of eigenstates and both observables can simultaniously be specified with infinite precision.

Example for commutator:
\([\hat{x},\hat{p}]\psi = i\hbar\psi\)
We cannot know position and momentum at the same time
\([\hat{a},\hat{a}] = 0 ,\forall \hat{a}\)
\([\frac{\hat{p}^2}{2m},\hat{p}] = 0\)
We can know kinetic energy and momentum at the same time.

Expectation value of an operator \(\hat{O}\) for a state \(\ket{\Psi}\)
 \(\braket{\hat{O} } \equiv \braket{\psi|\hat{O}|\psi}\) \(= \int O_\alpha |\psi(\alpha)|^2d\alpha\) 
With \(\ket{O_\alpha} \) the eigenstates of \(\hat{O}\)

Q: Schrödinger equation:
[...]
with[...]the [...] function.
A: \(i\hbar \dede{\Psi}{t} = -\frac{\hbar^2}{2m}\dede{^2\Psi}{x^2} + V\Psi\); \(V(x)\); potential energy
Back Extra: Note: Here we only consider conservative forces.

Q: The Hamiltonian (in QM):
\(\hat{H} = \) [...][...]
A: \(-\frac{\hbar^2}{2m}\dede{^2}{x^2} + \hat{V}\); \(= \frac{\hat{p}^2}{2m} + \hat{V}\)

Schrödinger Equation using the Hamiltonian:
\(i\hbar\dede{\Psi}{t} = \hat{H}\Psi\)
The eigenstates of \(\hat{H}\) lead to the time-independent Schrödinger Equation (TISE)
\(\hat{H}\Psi_n = E_n\Psi_n\)

Because the eigenvalues of \(\hat{H} \) can be used as a basis we can write the time dependent states as:
\(\Psi_n(x,t) = e^{-iE_nt/\hbar}\Psi_n(x)\)  with \(\Psi_n(x) \equiv \Psi_n(x,t=0)\)
Those energy eigenstates have the nice property that their probability density doesn't change in time

Q: To solve the S.E. we follow the following procedure:
1. [...]
2. [...]
3. The solution is given by [...]
A: Solve TISE for a given potential \(V(x)\) and an inintial state \(\ket{\Psi_0}\) (get \(\{\ket{\psi_n}\} \) and \(E_n\)); Write the initial state in the basis of the energy eigenstates \(\{\ket{\psi_n}\}\); \(\ket{\Psi(t)} = \sum_n c_ne^{-iE_nt/\hbar}\ket{\psi_n}\)

Q: Definition: [...]
The operator [...]
Measuring  such an eigenvalue will result [...]
A: Operator with degeneracy:; has multiple orthonormal eigenstates to the same eigenvalue.; not in a total collapse but only in a projection to a subspace of the wavefunction.

Q: Solution to [...]
[...]
A: the free particle in QM:; \(\Psi(x,t) = C_\pm e^{\pm ik(x\mp\frac{\hbar k}{2m}t)} \)

Q: Conditions to the wavefunction:
1. [...]
2. [...]
A: \(\Psi(x)\) is continuous; \(\dd{\Psi(x)}{x}\) is continuous except where \(V(x) \to \infty\)

Definition parity operator:
\(\hat{\Pi}\psi(y) = \psi(-y)\)
This operator commutes with \(\hat{H}\)
It's eigenvalues are \(\pm 1\)  which corresponds to even and odd functions