Let's break down and solve the problem step by step. --- ### **Q4. Two Non-Interacting Identical Particles in a Harmonic Oscillator** #### **Given:** - Two **identical particles** (so we must consider quantum statistics: Bose for integer spin, Fermi for half-integer spin). - Each particle is in a **common external harmonic oscillator**. - Find **ground state energy levels and wave functions** for: a) **Two spin-1 particles** (bosons) b) **Two spin-1/2 particles** (fermions) --- ## **Step 1: Single Particle in Harmonic Oscillator** - **Energy eigenvalues:** \( E_n = \left(n + \frac{1}{2}\right)\hbar \omega \), where \( n = 0, 1, 2, \ldots \) - **Ground state energy:** \( E_{gs} = \frac{1}{2}\hbar\omega \) --- ## **Step 2: System of Two Identical Particles** ### **Total Hamiltonian:** Since the particles are non-interacting, \[ H = H_1 + H_2 \] - **Total energy:** \( E_{n_1, n_2} = (n_1 + n_2 + 1)\hbar\omega \) - **Ground state:** both in \( n=0 \): \( E_{0,0} = \hbar\omega \) --- ## **Step 3: Symmetry of the Total Wavefunction** The **total wavefunction** is the product of the **spatial part** and the **spin part**. - **Bosons (integer spin):** Total wavefunction must be **symmetric** under particle exchange. - **Fermions (half-integer spin):** Total wavefunction must be **antisymmetric** under particle exchange. --- ## **a) Two Spin-1 Particles (Bosons)** - **Spin-1 particles are bosons.** - Both can occupy the same spatial state. ### **Spatial part:** - Both in ground state \( n_1 = n_2 = 0 \). \[ \psi_{spatial}^{gs}(x_1, x_2) = \psi_0(x_1)\psi_0(x_2) \] This is symmetric under exchange \( x_1 \leftrightarrow x_2 \). ### **Spin part:** - For two spin-1 particles, the combined spin states are: - **Symmetric:** total spin \( S = 2 \) (quintet), \( S = 1 \) (triplet) - **Antisymmetric:** total spin \( S = 0 \) (singlet) But since the spatial part is already symmetric, **spin part must also be symmetric** for the total wavefunction to be symmetric. **Allowed total spin states**: \( S=2 \) (quintet) and \( S=1 \) (triplet). ### **Energy:** \[ E_{gs} = \hbar\omega \] ### **Wavefunction:** \[ \Psi_{gs}(x_1, x_2; s_1, s_2) = \psi_0(x_1)\psi_0(x_2)\ \chi_{sym}(s_1, s_2) \] where \( \chi_{sym} \) is any symmetric spin state (total spin 2 or 1). --- ## **b) Two Spin-1/2 Particles (Fermions)** - **Spin-1/2 particles are fermions.** - **Pauli exclusion principle:** No two fermions can occupy the same quantum state. ### **Spatial part:** - If both are in ground state (\( n_1 = n_2 = 0 \)), the spatial part is symmetric. - But the total wavefunction must be **antisymmetric**. ### **Spin part:** - For two spin-1/2 particles: - **Symmetric spin state:** triplet (\( S=1 \)), 3 states - **Antisymmetric spin state:** singlet (\( S=0 \)), 1 state To get antisymmetric total wavefunction: - **If spatial part is symmetric (both in ground state):** Spin part must be **antisymmetric** (singlet). - **If spatial part is antisymmetric (one in \( n=0 \), one in \( n=1 \)):** Spin part can be symmetric (triplet). #### **But**: For ground state, both in \( n=0 \). So, spatial part is symmetric, so we need singlet spin state. ### **Energy:** \[ E_{gs} = \hbar\omega \] ### **Wavefunction:** \[ \Psi_{gs}(x_1, x_2; s_1, s_2) = \psi_0(x_1)\psi_0(x_2)\ \chi_{antisym}(s_1, s_2) \] where \( \chi_{antisym}(s_1, s_2) = \frac{1}{\sqrt{2}} \left( |\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle \right) \) (singlet state). --- ## **Summary Table** | Case | Energy of Ground State | Spatial Part | Spin Part | Allowed Spin States | |--------------------------|-----------------------|------------------------------|-----------------------------------|--------------------| | Two spin-1 (bosons) | \( \hbar\omega \) | \( \psi_0(x_1)\psi_0(x_2) \) | Symmetric | S=2, S=1 | | Two spin-1/2 (fermions) | \( \hbar\omega \) | \( \psi_0(x_1)\psi_0(x_2) \) | Antisymmetric (singlet) | S=0 | --- ### **Final Answers** ### **a) For two spin-1 particles:** - **Ground state energy:** \( \boxed{\hbar\omega} \) - **Wavefunction:** \[ \boxed{ \Psi_{gs}(x_1, x_2; s_1, s_2) = \psi_0(x_1)\psi_0(x_2)\ \chi_{sym}(s_1, s_2) } \] where \( \chi_{sym} \) is any symmetric two-spin-1 state. --- ### **b) For two spin-1/2 particles:** - **Ground state energy:** \( \boxed{\hbar\omega} \) - **Wavefunction:** \[ \boxed{ \Psi_{gs}(x_1, x_2; s_1, s_2) = \psi_0(x_1)\psi_0(x_2) \frac{1}{\sqrt{2}}\left( |\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle \right) } \] where the spin state is the singlet (antisymmetric under exchange). --- If you need more details or want the excited states, let me know!