MIT 8.04 - Quantum Mechanics I: Principles & Applications Flashcards

Question 1

Quantum Operators ($\hat{A}$)

Accepted Answer

Mathematical mappings acting on a wave function to yield physical observables. In quantum mechanics, every measurable physical quantity (position, momentum, energy) corresponds to a linear Hermitian operator.

Question 2

Wave Function ($\Psi$)

Accepted Answer

A complex-valued mathematical function $\Psi(x,t)$ that completely describes the quantum state of a physical system. The absolute square $|\Psi(x,t)|^2$ represents the probability density of finding the particle at position $x$ and time $t$.

Question 3

Born Interpretation (Normalization)

Accepted Answer

The physical postulate stating that $|\Psi(x,t)|^2 dx$ is the probability of finding a particle in the interval $[x, x+dx]$. Since the total probability must equal 1, every valid wave function must satisfy the normalization condition: $\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1$.

Question 4

Position and Momentum Operators

Accepted Answer

In spatial representation, the position operator is $\hat{x} = x$, and the linear momentum operator is $\hat{p} = -i\hbar \frac{\partial}{\partial x}$.

Question 5

Hermitian Operators

Accepted Answer

An operator $\hat{A}$ satisfying $\langle \psi | \hat{A} \phi \rangle = \langle \hat{A} \psi | \phi \rangle$. Hermitian operators guarantee real eigenvalues (measurable physical quantities) and orthogonal eigenfunctions corresponding to distinct eigenvalues.

Question 6

Eigenvalue Equation

Accepted Answer

An equation of the form $\hat{A} \psi_a = a \psi_a$, where $\hat{A}$ is an operator, $\psi_a$ is an eigenfunction (eigenstate), and $a$ is a scalar eigenvalue representing a possible outcome of measuring observable $A$.

Question 7

Expectation Value ($\langle A \rangle$)

Accepted Answer

The ensemble average outcome of repeated measurements of an observable $\hat{A}$ on identical quantum states $\Psi$: $\langle A \rangle = \int_{-\infty}^{\infty} \Psi^* \hat{A} \Psi dx$.

Question 8

Time-Dependent Schrödinger Equation (TDSE)

Accepted Answer

The fundamental partial differential equation governing quantum system dynamics: $i\hbar \frac{\partial}{\partial t}\Psi(x,t) = \hat{H}\Psi(x,t)$, where $\hat{H}$ is the Hamiltonian operator.

Question 9

Hamiltonian Operator ($\hat{H}$)

Accepted Answer

The operator corresponding to the total energy of a system, equal to the sum of kinetic and potential energy operators: $\hat{H} = \frac{\hat{p}^2}{2m} + V(x) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)$.

Question 10

Time-Independent Schrödinger Equation (TISE)

Accepted Answer

An eigenvalue equation obtained via separation of variables for time-independent potentials: $\hat{H}\psi(x) = E\psi(x)$, where $E$ represents the energy eigenvalue of the stationary state $\psi(x)$.

Question 11

Stationary States

Accepted Answer

Solutions to TDSE with time-independent potential of the form $\Psi(x,t) = \psi(x)e^{-iEt/\hbar}$. Their spatial probability densities $|\Psi(x,t)|^2 = |\psi(x)|^2$ do not change over time.

Question 12

Superposition Principle

Accepted Answer

If $\psi_1$ and $\psi_2$ are valid quantum states, any linear combination $\Psi = c_1\psi_1 + c_2\psi_2$ is also a valid quantum state. The probability of measuring the eigenvalue corresponding to state $n$ is $|c_n|^2$.

Question 13

Commutator ($[\hat{A}, \hat{B}]$)

Accepted Answer

An operation defined as $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$. If $[\hat{A}, \hat{B}] = 0$, the operators commute and share a simultaneous set of eigenfunctions (can be measured simultaneously with infinite precision).

Question 14

Canonical Commutant relation

Accepted Answer

The fundamental commutation relation between position and momentum operators: $[\hat{x}, \hat{p}] = i\hbar$.

Question 15

Heisenberg Uncertainty Principle

Accepted Answer

A fundamental limit on the precision with which certain pairs of physical quantities (like position and momentum) can be known simultaneously: $\Delta x \Delta p \ge \frac{\hbar}{2}$.

Question 16

Generalized Uncertainty Principle

Accepted Answer

For any two Hermitian operators $\hat{A}$ and $\hat{B}$, the product of their standard deviations is bounded by: $\Delta A \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|$.

Question 17

Particle in a 1D Infinite Square Well

Accepted Answer

A model system with potential $V(x) = 0$ for $0 \le x \le L$ and $V(x) = \infty$ elsewhere. Wave functions are $\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$, with quantized energy levels $E_n = \frac{n^2 \pi^2 \hbar^2}{2m L^2}$ for $n=1,2,3,...$

Question 18

Quantum Tunneling

Accepted Answer

A quantum phenomenon where a particle has a non-zero probability of passing through a potential energy barrier higher than its kinetic energy ($E < V_0$). The wave function decays exponentially inside the classically forbidden region.

Question 19

Quantum Harmonic Oscillator (QHO)

Accepted Answer

A system subject to a quadratic potential $V(x) = \frac{1}{2}m\omega^2 x^2$. Its energy eigenvalues are strictly quantized as $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$ for $n = 0, 1, 2, ...$

Question 20

Ladder Operators (QHO)

Accepted Answer

Operators $\hat{a}$ (lowering/annihilation) and $\hat{a}^\dagger$ (raising/creation) that decrease or increase the energy state of a QHO by $\hbar\omega$: $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$ and $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$.

Question 21

Zero-Point Energy

Accepted Answer

The lowest possible energy that a quantum mechanical system may possess (ground state energy). For the quantum harmonic oscillator, $E_0 = \frac{1}{2}\hbar\omega > 0$, a direct consequence of the uncertainty principle.

MIT 8.04 - Quantum Mechanics I: Principles & Applications

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