Let $|\psi_1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $|\psi_2\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ represent two possible states of a two-level quantum system. The state obtained by the incoherent superposition of $|\psi_1\rangle$ and $|\psi_2\rangle$ is given by a density matrix that is defined as $\rho ≡ c_1|\psi_1\rangle\langle\psi_1| + c_2|\psi_2\rangle\langle\psi_2|$. If $c_1 = 0.4$ and $c_2 = 0.6$, the matrix element $\rho_{22}$ (rounded off to one decimal place) is ________
Quantum States Definition
The two given quantum states are represented by column vectors:
- $|\psi_1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$
- $|\psi_2\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$
Density Matrix Formalism
The density matrix $\rho$ for an incoherent superposition is given by:
$\rho = c_1|\psi_1\rangle\langle\psi_1| + c_2|\psi_2\rangle\langle\psi_2|$
where $c_1 = 0.4$ and $c_2 = 0.6$.
Calculating Projector Operators
First, calculate the outer products $|\psi_1\rangle\langle\psi_1|$ and $|\psi_2\rangle\langle\psi_2|$:
- $|\psi_1\rangle\langle\psi_1| = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$
- $|\psi_2\rangle\langle\psi_2| = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$
Constructing the Density Matrix
Substitute these into the density matrix formula:
$\rho = 0.4 \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + 0.6 \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$
$\rho = \begin{pmatrix} 0.4 \times 1 & 0.4 \times 0 \\ 0.4 \times 0 & 0.4 \times 0 \end{pmatrix} + \begin{pmatrix} 0.6 \times 0 & 0.6 \times 0 \\ 0.6 \times 0 & 0.6 \times 1 \end{pmatrix}$
$\rho = \begin{pmatrix} 0.4 & 0 \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & 0.6 \end{pmatrix}$
$\rho = \begin{pmatrix} 0.4 & 0 \\ 0 & 0.6 \end{pmatrix}$
Finding the Matrix Element $\rho_{22}$
The matrix element $\rho_{22}$ is the element in the second row and second column of the density matrix $\rho$. From the calculated matrix:
$\rho_{22} = 0.6$
Rounding off to one decimal place, the value remains 0.6.
Important Questions from Operators Commutators Heisenberg Picture
The wavefunction of a particle in one dimension is given by
$\psi(x) = \begin{cases} M, & -a < x < a \\ 0, & \text{otherwise.} \end{cases}$
Here $M$ and $a$ are positive constants. If $\phi(p)$ is the corresponding momentum space wavefunction, which one of the following plots best represents $|\phi(p)|^2$ ?- From the pairs of operators given below, identify the ones which commute. Here $l$ and $j$ correspond to the orbital angular momentum and the total angular momentum, respectively.
- An electromagnetic pulse has a pulse width of $10^{-3}$ s. The uncertainty in the momentum of the corresponding photon is of the order of $10^{-N}$ kg m $s^{-1}$, where $N$ is an integer. The value of $N$ is ________ (speed of light = $3 \times 10^8$ m $s^{-1}$, h = $6.6 \times 10^{-34}$ J s)
- In cylindrical coordinates $(s, \varphi, z)$, which of the following is a Hermitian operator?
- Let $|m\rangle$ and $|n\rangle$ denote the energy eigenstates of a one-dimensional simple harmonic oscillator. The position and momentum operators are $\hat{X}$ and $\hat{P}$, respectively. The matrix element $\langle m|\hat{P}\hat{X}|n\rangle$ is non-zero when
