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Chapter 3: Q14P (page 113)

Prove the famous "(your name) uncertainty principle," relating the uncertainty in positionA=x to the uncertainty in energyB=p2/2m+v:

蟽x蟽H鈮�h2m|p|

For stationary states this doesn't tell you much-why not?

Short Answer

Expert verified

The uncertainty principle is蟽x蟽H鈮�h2mp

Step by step solution

01

Concept used

The generalized uncertainty principle for two observables A and B is given by:

蟽A2蟽B2鈮�12iA^,B^2

02

Calculate the uncertainty principle

The generalized uncertainty principle for two observables A and B is given by:

蟽A2蟽B2鈮�12iA^,B^2

The position-energy uncertainty relation is:

蟽x2蟽H2鈮�12ix^,H^2 ....(1)

So, we need to find the commutatorx^,H^as:

X^,H^g=-h22m脳鈭�2g鈭�X2+xVg+-h22m鈭�2鈭�x2xg-xVg=h22m-x鈭�2g鈭�X2+2鈭�g鈭�x+x鈭�2g鈭�X2=h2m鈭�g鈭�x=ihmpg

Substitute in equation 1:

蟽x2蟽H2鈮�12ix^,H^]2=h24m2p2

So, the uncertainty principle here becomes

蟽x蟽H鈮�h2mp

For stationary states, this doesn't tell you much because the average position of the particle doesn't change, 蟽H=0andp=0.

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Most popular questions from this chapter

Consider the operator Q^=d2/d蠒2, where (as in Example 3.1)蠒 is the azimuthal angle in polar coordinates, and the functions are subject to Equation 3.26. Is Q^Hermitian? Find its eigenfunctions and eigenvalues. What is the spectrum of Q^? Is the spectrum degenerate?

(a) For what range of vis the function f(x)=x''in Hilbert space, on the interval (0.1)? Assume vis real, but not necessarily positive.

(b) For the specific case v=1/2, is f(x)in this Hilbert space? What aboutxf(x)? How about (d/dx)f(x)?

Show that the energy-time uncertainty principle reduces to the "your name" uncertainty principle (Problem 3.14), when the observable in question is x.

(a) Write down the time-dependent "Schr枚dinger equation" in momentum space, for a free particle, and solve it. Answer: exp(-ip2t/2mh)桅(p,0).

(b) Find role="math" localid="1656051039815" 桅(p,0)for the traveling gaussian wave packet (Problem 2.43), and construct 桅(p,t)for this case. Also construct |桅(p,t)|2, and note that it is independent of time.

(c) Calculaterole="math" localid="1656051188971" pandrole="math" localid="1656051181044" p2by evaluating the appropriate integrals involving桅, and compare your answers to Problem 2.43.

(d) Show thatrole="math" localid="1656051421703" <H>=<p>2/2m+<H>0(where the subscript denotes the stationary gaussian), and comment on this result.

Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if P^and Q^have a complete set of common eigenfunctions, then[P^路Q^]f=0 for any function in Hilbert space.
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