PH5170 Quiz 1-From basic quantum mechanics to spin Solution

Quiz I
From basic quantum mechanics to spin
1. Absence of degenerate bound states in one spatial dimension: Two or more quantum states are said

to be degenerate if they are described by distinct solutions to the time-independent Schrodinger equation corresponding to the same energy. For example, the free particle states are doubly degenerate—one solution describes motion to the right and the other motion to the left. It is not an accident that we have not encountered normalizable degenerate solutions in one spatial dimension. By following the steps listed below, prove that there are no degenerate bound states in one spatial dimension.
(a) Suppose that there are two solutions, say, ψ1 and ψ2, with the same energy E. Using the time-independent Schrodinger equation, show that the quantity
dψ dψ1
W
2. The translation operator: Recall that a wavefunction can be translated by the vector a in three

dimensions using the operator
,
where pˆ denotes the momentum operator. Let |ψ be a state vector describing a system and, upon translation, let
|ψ → |ψ¯ = Tˆ(a)|ψ.
3. Commuting operators and simultaneous eigenkets: Consider a three-dimensional ket space. If a

certain set of orthonormal kets, say,ˆ are found to be represented by the matrices|1, |2 and |3, are used as the base kets, the operators Aˆ and
B

with a and b both real.
(d) If the Aˆ and Bˆ represent observables and, if they can be measured simultaneously, construct a new set of orthonormal kets which are simultaneous eigenkets of both the operators. What
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4. Wave function describing the coherent state: Earlier, that we had arrived at the wave function ψ0(x) = x|0 describing the ground state of the oscillator using the following definition of the ground state |0: aˆ|0 = 0
and the position representation of the lowering operator ˆa. Recall that the coherent state |α is defined through the relation aˆ|α = α|α,
where α is a complex number.
5. Maximizing uncertainty: Recall that, according to the generalized uncertainty principle, the uncer-

tainty associated with two observables represented by the operators Aˆ and Bˆ is given by
,
where the uncertainty, say, ΔA2, is defined as ΔA2 = Aˆ2−Aˆ2 and the angular brackets as usual represent expectation values evaluated in a given state. Let, as usual, |+ and |− represent the eigenkets of the Sˆz operator describing a spin- system.
ΔSx2ΔSy2.
Note: It will be convenient to express the general stated describing the spin- system as follows: |χ = cos(θ/2)|+ + sin(θ/2)eiφ |−.
(b) Verify explicitly that, for the state |χ you have found, the uncertainty relation for the operators

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