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Consider a particle moving in a 2D infinite rectangular well defined by V = 0 for 0 < x < L₁ and 0 ≤ y ≤ L2, and V = ∞ elsewhere. Outside the well, the wavefunction (x, y) is zero. Inside the well, the wavefunction (x, y) obeys the standing wave condition in the x and y direction, so it is given as: where A is a constant. (x, y) = A sin(k₁x) sin(k₂y), (1) (a) The wavenumber k₁ in the x direction is quantized in terms of an integer n₁. Using the standing wave condition, find the possible values of k₁. (b) The wavenumber k2 in the y direction is quantized in terms of a different integer n₂. Using the standing wave condition, find the possible values of k₂. (c) Each state of the 2D infinite rectangular well is defined by the pair of quantum numbers (n₁, n₂). What is the energy of the state Enna?

Consider a particle moving in a 2D infinite rectangular well defined by V = 0 for 0 < x < L₁ and 0 ≤ y ≤ L2, and V = ∞ elsewhere. Outside the well, the wavefunction (x, y) is zero. Inside the well, the wavefunction (x, y) obeys the standing wave condition in the x and y direction, so it is given as: where A is a constant. (x, y) = A sin(k₁x) sin(k₂y), (1) (a) The wavenumber k₁ in the x direction is quantized in terms of an integer n₁. Using the standing wave condition, find the possible values of k₁. (b) The wavenumber k2 in the y direction is quantized in terms of a different integer n₂. Using the standing wave condition, find the possible values of k₂. (c) Each state of the 2D infinite rectangular well is defined by the pair of quantum numbers (n₁, n₂). What is the energy of the state Enna?

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Consider a particle moving in a 2D infinite rectangular well defined by \( V = 0 \) for \( 0 < x < L_1 \) and \( 0 < y < L_2 \), and \( V = \infty \) elsewhere. Outside the well, the wavefunction \( \psi(x, y) \) is zero. Inside the well, the wavefunction \( \psi(x, y) \) obeys the standing wave condition in the \( x \) and \( y \) direction, so it is given as: \[ \psi(x, y) = A \sin(k_1 x) \sin(k_2 y), \] where \( A \) is a constant. (a) The wavenumber \( k_1 \) in the \( x \) direction is quantized in terms of an integer \( n_1 \). Using the standing wave condition, find the possible values of \( k_1 \). (b) The wavenumber \( k_2 \) in the \( y \) direction is quantized in terms of a different integer \( n_2 \). Using the standing wave condition, find the possible values of \( k_2 \). (c) Each state of the 2D infinite rectangular well is defined by the pair of quantum numbers \( (n_1, n_2) \). What is the energy of the state \( E_{n_1, n_2} \)? (1) what are the steps for determining these values and energy based on the boundary conditions?
Transcribed Image Text:Consider a particle moving in a 2D infinite rectangular well defined by \( V = 0 \) for \( 0 < x < L_1 \) and \( 0 < y < L_2 \), and \( V = \infty \) elsewhere. Outside the well, the wavefunction \( \psi(x, y) \) is zero. Inside the well, the wavefunction \( \psi(x, y) \) obeys the standing wave condition in the \( x \) and \( y \) direction, so it is given as: \[ \psi(x, y) = A \sin(k_1 x) \sin(k_2 y), \] where \( A \) is a constant. (a) The wavenumber \( k_1 \) in the \( x \) direction is quantized in terms of an integer \( n_1 \). Using the standing wave condition, find the possible values of \( k_1 \). (b) The wavenumber \( k_2 \) in the \( y \) direction is quantized in terms of a different integer \( n_2 \). Using the standing wave condition, find the possible values of \( k_2 \). (c) Each state of the 2D infinite rectangular well is defined by the pair of quantum numbers \( (n_1, n_2) \). What is the energy of the state \( E_{n_1, n_2} \)? (1) what are the steps for determining these values and energy based on the boundary conditions?
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