Consider two electrons in a one-dimensional infinite square length of width L. Assume the two electrons are in the same spin state (x1 = X2). (a) Find the wavefunction and energy of the ground state of this two-electron state. (b) Find the energy and wavefunction of the first excited state, still assuming the electrons are in the same spin state, X1 = X2

An electron is trapped in an infinitely deep one-dimensional well of width 0,251 nm. Initially the electron occupies the n=4 state. Find the energies of other photons that might be emitted if the electron takes other paths between the n=4 state and the ground state

Complete the derivation of E = Taking the derivatives we find (Use the following as necessary: k₁, K₂ K3, and 4.) +- ( ²) (²) v² = SO - #2² - = 2m so the Schrödinger equation becomes (Use the following as necessary: K₁, K₂, K3, ħ, m and p.) 亢 2mm(K² +K ² + K² v k₁ = E = = EU The quantum numbers n, are related to k, by (Use the following as necessary: n, and L₁.) лħ n₂ π²h² 2m √2m h²²/0₁ 2m X + + by substituting the wave function (x, y, z) = A sin(kx) sin(k₂y) sin(kz) into - 13³3). X What is the origin of the three quantum numbers? O the Schrödinger equation O the Pauli exclusion principle O the uncertainty principle Ⓒthe three boundary conditions 2² 7²4 = E4. 2m

Particle of mass m moves in a three-dimensional box with edge lengths L1, L2, and L3. (a) Find the energies of the six lowest states if L1 =L, L2 = 2L, and L3 = 2L. (b) Which if these energies are degenerate?

For an infinite potential well of length L, determine the difference in probability that a particle might be found between x = 0.25L and x = 0.75L between the n = 3 state and the n = 5 states.

a 4. 00, -Vo, V(z) = 16a 0, Use the WKB approximation to determine the minimum value that V must have in order for this potential to allow for a bound state.

The wave function for the ground state of hydrogen is given by 100(0,0) = Ae¯¯r/ª Find the constant A that will normalize this wave func- tion over all space.

An electron in a hydrogen atom is approximated by a one-dimensional infinite square well potential. The normalised wavefunction of an electron in a stationary state is defined as *(x) = √√ sin (""). L where n is the principal quantum number and L is the width of the potential. The width of the potential is L = 1 x 10-¹0 m. (a) Explain the meaning of the term normalised wavefunction and why normalisation is important. (b) Use the wavefunction defined above with n = 2 to determine the probability that an electron in the first excited state will be found in the range between x = 0 and x = 1 × 10-¹¹ m. Use an appropriate trigonometric identity to simplify your calculation. (c) Use the time-independent Schrödinger Equation and the wavefunction defined above to find the energies of the first two stationary states. You may assume that the electron is trapped in a potential defined as V(x) = 0 for 0≤x≤L ∞ for elsewhere.

We have a three dimensional vector space where |P1), |P2) and |23) form a complete orthonormal basis. In this vector space we have two states |a)=5i|1)+3i 2)+(-2+2i) 3) and |B) =4i|1)-5 i) Calculate (a and (B, in terms of the dual basis vectors (y|, (p2|, (P3|. ii) Calculate the inner/scalar products (alB) and (Ba). Show that (8|a) =(a|B)".

Explain each step