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1. Note: in this question we do not use natural units so c and h are not equal to one. Consider a ring of N identical balls, all of mass m. In equilibrium, balls lie equally spread around the ring with a distance a between nearest neighbours. The Hamil- tonian operator for the normal modes may be written as mu (1) [2m Wavenumbers are labelled k, p. q etc. Sums over wavenumbers, such as E are taken over all allowed wavenumbers and are symmetric (that is if a value k is present in the sum then the value -k is also in the sum). You need not consider the k = 0 mode explicitly. For a normal mode of wave number k, U, is the hermitian position operator for that mode (with units of length), and P is the hermitian momentum operator (with units of momentum). The dispersion relation is we = IV4w sin (ka/2) +° where w and 2 are both fixed characteristic frequencies. The operators U, and P, satisfy 0 = 0. P = P... [0,. Pal = ihőp-q. [0p. O = 0. [P, Pel = 0. (2) ) Show that l = (h/mw.) has units of length. Annihilation operators may be defined as mw 2n (3) What are the units of the annihilation operator? Prove that this annihilation operator and its hermitian conjugate satisfy the commutation relations [â, a = 6p.a. (âp. âg] = 0 and [a. a = 0.

1. Note: in this question we do not use natural units so c and h are not equal to one. Consider a ring of N identical balls, all of mass m. In equilibrium, balls lie equally spread around the ring with a distance a between nearest neighbours. The Hamil- tonian operator for the normal modes may be written as mu (1) [2m Wavenumbers are labelled k, p. q etc. Sums over wavenumbers, such as E are taken over all allowed wavenumbers and are symmetric (that is if a value k is present in the sum then the value -k is also in the sum). You need not consider the k = 0 mode explicitly. For a normal mode of wave number k, U, is the hermitian position operator for that mode (with units of length), and P is the hermitian momentum operator (with units of momentum). The dispersion relation is we = IV4w sin (ka/2) +° where w and 2 are both fixed characteristic frequencies. The operators U, and P, satisfy 0 = 0. P = P... [0,. Pal = ihőp-q. [0p. O = 0. [P, Pel = 0. (2) ) Show that l = (h/mw.) has units of length. Annihilation operators may be defined as mw 2n (3) What are the units of the annihilation operator? Prove that this annihilation operator and its hermitian conjugate satisfy the commutation relations [â, a = 6p.a. (âp. âg] = 0 and [a. a = 0.

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This is my question for preparing the excam. I have exam tomorrow please give me step by step explanation for this question. I would highhly appreciate your help, Thanks!

1. Note: in this question we do not use natural units so c and h are not equal to one. Consider a ring of N identical balls, all of mass m. In equilibrium, balls lie equally spread around the ring with a distance a between nearest neighbours. The Hamil- tonian operator for the normal modes may be written as A=E2m (1) [2m Wavenumbers are labelled k, p, q etc. Sums over wavenumbers, such as . are taken over all allowed wavenumbers and are symmetric (that is if a value k is present in the sum then the value -k is also in the sum). You need not consider the k = 0 mode explicitly. For a normal mode of wave number k, On is the hermitian position operator for that mode (with units of length), and P is the hermitian momentum operator (with units of momentum). The dispersion relation is w = IV4w? sin (ka/2) +2| where w and S are both fixed characteristic frequencies. The operators U, and P satisfy 0 = 0. P = P. (0p, Pal = ihőp -a. 0p. Oal = 0, [Pp, Pal = 0. (2) (i) Show that & = (h/ mw.)2 has units of length. Annihilation operators may be defined as mw (3) 2n What are the units of the annihilation operator? Prove that this annihilation operator and its hermitian conjugate satisfy the commutation relations [â, a = 6pa. (ập, âg] = 0 and [3, a) = 0. (ii) Show that the Hamiltonian A of (1) may be rewritten as Hint: you may start from (4) and work towards (1). (i) The Hamiltonian for a single scalar relativistic field o in one spatial dimension with conjugate momentum operator n is dx x))* + c? Əộ(t, x) ax -($(t. x) By writing (t,x) = (2m)- S dk ek(t, k), show how to match the o terms with the Û terms of (1) in the limit a0. You may ignore the n and P terms. As part of your answer you must express c and M from (5) in terms of a, w and 2 of (1) such that the dimensions are compatible.
Transcribed Image Text:1. Note: in this question we do not use natural units so c and h are not equal to one. Consider a ring of N identical balls, all of mass m. In equilibrium, balls lie equally spread around the ring with a distance a between nearest neighbours. The Hamil- tonian operator for the normal modes may be written as A=E2m (1) [2m Wavenumbers are labelled k, p, q etc. Sums over wavenumbers, such as . are taken over all allowed wavenumbers and are symmetric (that is if a value k is present in the sum then the value -k is also in the sum). You need not consider the k = 0 mode explicitly. For a normal mode of wave number k, On is the hermitian position operator for that mode (with units of length), and P is the hermitian momentum operator (with units of momentum). The dispersion relation is w = IV4w? sin (ka/2) +2| where w and S are both fixed characteristic frequencies. The operators U, and P satisfy 0 = 0. P = P. (0p, Pal = ihőp -a. 0p. Oal = 0, [Pp, Pal = 0. (2) (i) Show that & = (h/ mw.)2 has units of length. Annihilation operators may be defined as mw (3) 2n What are the units of the annihilation operator? Prove that this annihilation operator and its hermitian conjugate satisfy the commutation relations [â, a = 6pa. (ập, âg] = 0 and [3, a) = 0. (ii) Show that the Hamiltonian A of (1) may be rewritten as Hint: you may start from (4) and work towards (1). (i) The Hamiltonian for a single scalar relativistic field o in one spatial dimension with conjugate momentum operator n is dx x))* + c? Əộ(t, x) ax -($(t. x) By writing (t,x) = (2m)- S dk ek(t, k), show how to match the o terms with the Û terms of (1) in the limit a0. You may ignore the n and P terms. As part of your answer you must express c and M from (5) in terms of a, w and 2 of (1) such that the dimensions are compatible.
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