L i In 1D, the commutator of the position and velocity operators is [, vx] m...

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L iħ In 1D, the commutator of the position and velocity operators is [, vx] m Expressing your answer in operators, numbers, and constants only, determine [î, û] B. Consider a particle, mass m, in a 2D box, where V (x, y) =0 for 0 ≤x≤ a and 0 ≤ y≤ 2a (Note that this box is not a square.) It is known that stationary states of this potential have the form Un,ny = X(x)Y(y), where X and Y are solutions of the related 1D problem, and differ only in their co-ordinates, independent quantum numbers, and the distinct lengths of the box in the two dimensions. L [1 mark] Using what you know about the allowed energies of the 1D PIB, write down an expression for the allowed energies of this 2D problem (Enny), in terms of n, ny, numbers, constants, and the length parameter a. ■ [1 mark] Consider the family of functions, Y (y), that are components of the overall wavefunctions for stationary states. Sketch the version of this (1D) function that would contribute the smallest amount of energy to any overall wavefunction. Label any points on the y-axis where Y(y) = 0 (There is no need to specify the height of the function.) Hint: The horizontal axis in this plot should be y, and the vertical axis the values of Y (y) ■ L iħ In 1D, the commutator of the position and velocity operators is [, vx] m Expressing your answer in operators, numbers, and constants only, determine [î, û] B. Consider a particle, mass m, in a 2D box, where V (x, y) =0 for 0 ≤x≤ a and 0 ≤ y≤ 2a (Note that this box is not a square.) It is known that stationary states of this potential have the form Un,ny = X(x)Y(y), where X and Y are solutions of the related 1D problem, and differ only in their co-ordinates, independent quantum numbers, and the distinct lengths of the box in the two dimensions. L [1 mark] Using what you know about the allowed energies of the 1D PIB, write down an expression for the allowed energies of this 2D problem (Enny), in terms of n, ny, numbers, constants, and the length parameter a. ■ [1 mark] Consider the family of functions, Y (y), that are components of the overall wavefunctions for stationary states. Sketch the version of this (1D) function that would contribute the smallest amount of energy to any overall wavefunction. Label any points on the y-axis where Y(y) = 0 (There is no need to specify the height of the function.) Hint: The horizontal axis in this plot should be y, and the vertical axis the values of Y (y) ■