Chapter 2 Quiz
Instructions: Answer each question, then reveal the hidden answer to check your understanding. Aim for at least 16/20 before moving to Chapter 3.
Q1. What physical quantity does $|\psi(x,t)|^2$ represent?
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$|\psi(x,t)|^2$ is the **probability density** — the probability per unit length of finding the particle at position $x$ at time $t$. To get an actual probability, you must integrate over a region: $P(a \leq x \leq b) = \int_a^b |\psi|^2\,dx$.Q2. Write the time-dependent Schrödinger equation for a particle of mass $m$ in a potential $V(x)$.
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$$i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi$$ Or in compact notation: $i\hbar\,\partial_t\psi = \hat{H}\psi$.Q3. The TDSE is first-order in time. Why does this mean the initial condition $\psi(x,0)$ completely determines the future evolution, with no need to specify $\partial\psi/\partial t|_{t=0}$?
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A first-order ODE/PDE in time requires only one initial condition (the function at $t=0$). The Schrödinger equation itself determines $\partial\psi/\partial t$ from $\psi$, so specifying $\psi(x,0)$ is sufficient. This contrasts with Newton's law ($F=ma$, second order), which requires both position and velocity at $t=0$.Q4. What is the normalization condition for a one-dimensional wave function, and what is its physical meaning?
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$$\int_{-\infty}^{\infty}|\psi(x,t)|^2\,dx = 1$$ Physical meaning: the particle must be found *somewhere* with certainty. The total probability over all space equals 1.Q5. What is the momentum operator in the position representation?
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$$\hat{p} = -i\hbar\frac{\partial}{\partial x}$$Q6. Write the formula for the expectation value of an observable $A$ for a system in state $\psi$.
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$$\langle A \rangle = \int_{-\infty}^{\infty}\psi^*(x,t)\,\hat{A}\,\psi(x,t)\,dx$$ where $\hat{A}$ is the operator corresponding to observable $A$.Q7. What is a stationary state? Why is it called "stationary" despite having a time-dependent phase factor?
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A stationary state has the form $\psi(x,t) = \phi(x)e^{-iEt/\hbar}$, where $\phi$ satisfies $\hat{H}\phi = E\phi$. It is called "stationary" because the probability density $|\psi(x,t)|^2 = |\phi(x)|^2$ is independent of time. All measurable quantities (expectation values of time-independent operators) are constant. The phase factor $e^{-iEt/\hbar}$ is physically unobservable for a single stationary state.Q8. If a particle is in a stationary state with energy $E$, and you measure the energy, what result do you get? With what probability?
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You get exactly $E$ with probability 1. A stationary state is an eigenstate of $\hat{H}$ with eigenvalue $E$, so the energy is definite (zero variance).Q9. A particle is in the superposition $\psi = \frac{1}{\sqrt{3}}\phi_1 + \sqrt{\frac{2}{3}}\,\phi_2$, where $\phi_1$ and $\phi_2$ are orthonormal eigenstates of $\hat{H}$ with energies $E_1$ and $E_2$. What is $\langle E \rangle$?
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$$\langle E \rangle = |c_1|^2 E_1 + |c_2|^2 E_2 = \frac{1}{3}E_1 + \frac{2}{3}E_2$$Q10. In Question 9, if you measure the energy and get $E_1$, what is the state immediately after the measurement?
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The state collapses to $\phi_1$. The superposition is destroyed, and the system is now in a definite energy eigenstate.Q11. True or false: If two wave functions $\psi_1$ and $\psi_2$ are both solutions of the TDSE for the same Hamiltonian, then $\psi_1 + \psi_2$ is also a solution.
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**True.** This follows from the linearity of the Schrödinger equation (the superposition principle).Q12. What is the probability current density? Write its formula and state the continuity equation.
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$$j(x,t) = \frac{\hbar}{2mi}\left(\psi^*\frac{\partial\psi}{\partial x} - \frac{\partial\psi^*}{\partial x}\psi\right) = \frac{\hbar}{m}\operatorname{Im}\left(\psi^*\frac{\partial\psi}{\partial x}\right)$$ Continuity equation: $\frac{\partial|\psi|^2}{\partial t} + \frac{\partial j}{\partial x} = 0$. This expresses conservation of probability.Q13. List three requirements for a physically acceptable wave function.
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Any three of: 1. **Square-integrable** (normalizable): $\int|\psi|^2\,dx < \infty$ 2. **Continuous** everywhere 3. **Continuous first derivative** (wherever $V$ is finite) 4. **Single-valued**Q14. Why does imposing boundary conditions on the wave function lead to energy quantization for bound states?
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For bound states, $\psi$ must vanish at $x \to \pm\infty$ (normalizability). The TISE has oscillatory solutions where $E > V$ and exponential solutions where $E < V$. These must match smoothly at the boundaries between regions, which is only possible for specific discrete values of $E$. The boundary conditions act as a "filter" that selects the allowed energies.Q15. What is the difference between the time-dependent and time-independent Schrödinger equations? When can we use the time-independent form?
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The **TDSE** ($i\hbar\partial_t\psi = \hat{H}\psi$) governs the full time evolution of any quantum state. The **TISE** ($\hat{H}\phi = E\phi$) is an eigenvalue equation that arises from separation of variables when $V$ is time-independent. We can use the TISE to find stationary states (energy eigenstates), and then construct general solutions as superpositions of stationary states. The TISE cannot be used if $V$ depends on time.Q16. A Gaussian wave function $\psi(x) = (2\alpha/\pi)^{1/4}e^{-\alpha x^2}$ has uncertainty product $\sigma_x\sigma_p = \hbar/2$. What is special about this value?
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$\hbar/2$ is the **minimum** allowed by the Heisenberg uncertainty principle ($\sigma_x\sigma_p \geq \hbar/2$). The Gaussian is a **minimum-uncertainty state** — it saturates the lower bound. No wave function can have a smaller uncertainty product.Q17. Explain why $|\psi|^2$ is time-independent for a stationary state but time-dependent for a superposition of stationary states with different energies.
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For a single stationary state, $|\psi|^2 = |\phi|^2|e^{-iEt/\hbar}|^2 = |\phi|^2$ — the time-dependent phase cancels. For a superposition $\psi = c_1\phi_1 e^{-iE_1 t/\hbar} + c_2\phi_2 e^{-iE_2 t/\hbar}$, the cross terms in $|\psi|^2$ involve $e^{i(E_1-E_2)t/\hbar}$, which oscillates with frequency $(E_2-E_1)/\hbar$. Different energies $\Rightarrow$ different phases $\Rightarrow$ time-dependent interference $\Rightarrow$ time-dependent $|\psi|^2$.Q18. What does it mean to say the wave function is a "probability amplitude"?
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$\psi$ is not itself a probability — it is a complex number whose modulus squared gives the probability density. Just as a classical amplitude $A$ determines intensity $I \propto A^2$, the quantum amplitude $\psi$ determines probability density $|\psi|^2$. Crucially, amplitudes (not probabilities) are what add in quantum superposition, which is why interference occurs.Q19. Can the expectation value $\langle E \rangle$ for a state $\psi = c_1\phi_1 + c_2\phi_2$ ever equal an energy eigenvalue?
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Yes, but only in special cases. $\langle E \rangle = |c_1|^2 E_1 + |c_2|^2 E_2$. This equals $E_1$ only if $|c_2|^2 = 0$ (i.e., the state is just $\phi_1$), or equals $E_2$ only if $|c_1|^2 = 0$. For a genuine superposition ($c_1 \neq 0, c_2 \neq 0$), $\langle E \rangle$ is strictly between $E_1$ and $E_2$ and does not equal either eigenvalue. However, each individual measurement still yields either $E_1$ or $E_2$ — never $\langle E \rangle$ itself.Q20. In what sense is the Schrödinger equation "deterministic"? If it is deterministic, where does quantum randomness come from?