In this problem you derive the wavefunction ⟨𝐱 |𝐩⟩=e^ip ·𝐱 / ħ of a state of well-defined moment

step 1 For a photon, it has zero mass, this massless, so it is not appropriate to use p equals to mv to

In this problem you derive the wavefunction ⟨𝐱 |𝐩⟩=e^ip ·𝐱 /...

In this problem you derive the wavefunction $$ \langle\mathbf{x} \mid \mathbf{p}\rangle=\mathrm{e}^{\mathrm{ip} \cdot \mathbf{x} / \hbar} $$ of a state of well-defined momentum from the properties of the translation operator $U(\mathbf{a}) .$ The state $|\mathbf{k}\rangle$ is one of well-defined momentum $\hbar \mathbf{k}$. How would you characterise the state $\left|\mathbf{k}^{\prime}\right\rangle \equiv U(\mathbf{a})|\mathbf{k}\rangle ?$ Show that the wavefunctions of these states are related by $u_{\mathbf{k}^{\prime}}(\mathbf{x})=\mathrm{e}^{-\mathrm{ia} \cdot \mathbf{k}} u_{\mathbf{k}}(\mathbf{x})$ and $u_{\mathbf{k}^{\prime}}(\mathbf{x})=u_{\mathbf{k}}(\mathbf{x}-\mathbf{a}) .$ Hence obtain equation $(4.76)$

In this problem you derive the wavefunction
$$
\langle\mathbf{x} \mid \mathbf{p}\rangle=\mathrm{e}^{\mathrm{ip} \cdot \mathbf{x} / \hbar}
$$
of a state of well-defined momentum from the properties of the translation operator $U(\mathbf{a}) .$ The state $|\mathbf{k}\rangle$ is one of well-defined momentum $\hbar \mathbf{k}$. How would you characterise the state $\left|\mathbf{k}^{\prime}\right\rangle \equiv U(\mathbf{a})|\mathbf{k}\rangle ?$ Show that the wavefunctions of these states are related by $u_{\mathbf{k}^{\prime}}(\mathbf{x})=\mathrm{e}^{-\mathrm{ia} \cdot \mathbf{k}} u_{\mathbf{k}}(\mathbf{x})$ and $u_{\mathbf{k}^{\prime}}(\mathbf{x})=u_{\mathbf{k}}(\mathbf{x}-\mathbf{a}) .$ Hence obtain equation $(4.76)$

Key Concepts

Translation Operator

The translation operator is a unitary operator that shifts the spatial argument of a wavefunction by a constant displacement. In quantum mechanics, its action on a state corresponds to translating the entire system in space without altering its intrinsic physical properties. It plays a crucial role in connecting measurements in different reference frames and establishing the relationship between position and momentum representations.

Momentum Eigenstates

Momentum eigenstates are quantum states that are eigenfunctions of the momentum operator, associated with well-defined momenta. In the position representation, these states typically appear as plane waves. Their properties, such as invariance up to a phase factor under translations, reflect the fundamental commutation relations between position and momentum, which underpin much of quantum mechanics.

Wavefunction Representation

The wavefunction provides a complete description of a quantum state in the position space. It relates abstract quantum states to measurable probabilities through its modulus squared. In problems involving translations, understanding how the wavefunction changes under spatial shifts is essential, as it connects the mathematical formalism to observable effects such as interference and momentum conservation.

Phase Factors

Phase factors in quantum mechanics are multiplicative complex constants that do not affect the probability densities but are essential for maintaining the consistency of the theory under transformations. When a state is translated, the resulting phase factor is a direct consequence of the momentum eigenvalue, reflecting the deep connection between translation symmetry and conservation of momentum. This phase factor is crucial for ensuring the proper transformation behavior of wavefunctions.

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