Worksheet: The Schrödinger Wave Equation

In this worksheet, we will practice representing the properties of quantum particles with probability wave functions using the Schrödinger wave equation.

Q1:

A wave function is evaluated at the rectangular coordinates (๐‘ฅ,๐‘ฆ,๐‘ง)=(2,1,1) in arbitrary units. What are the spherical coordinates of this position?

  • A ๏€ป โˆš 9 , 6 2 , 1 5 ๏‡ โˆ˜ โˆ˜
  • B ๏€ป โˆš 5 , 6 2 , 2 9 ๏‡ โˆ˜ โˆ˜
  • C ( 6 , 5 2 , 1 5 ) โˆ˜ โˆ˜
  • D ๏€ป โˆš 7 , 6 9 , 2 4 ๏‡ โˆ˜ โˆ˜
  • E ๏€ป โˆš 6 , 6 6 , 2 7 ๏‡ โˆ˜ โˆ˜

Q2:

A wave function of a particle with mass ๐‘š is given by ฮจ(๐‘ฅ)=๏ฑ๐›ผ๐‘ฅโˆ’๐œ‹2๐›ผโ‰ค๐‘ฅโ‰ค๐œ‹2๐›ผ0,cosifotherwise where ๐›ผ=1.00ร—10๏Šง๏Šฆ๏Šฑ๏Šงm.

Find the probability that the particle can be found in the interval (0.00โ‰ค๐‘ฅโ‰ค0.50ร—10)๏Šฑ๏Šง๏Šฆ m.

Q3:

Given the complex-valued function ๐‘“(๐‘ฅ,๐‘ฆ)=2๐‘ฅโˆ’3๐‘–๐‘ฆ2๐‘ฅ+3๐‘–๐‘ฆ, calculate |๐‘“(๐‘ฅ,๐‘ฆ)|๏Šจ.

Q4:

Which of the following is the expectation value of the position squared for a particle that is in its ground state in a box of length ๐ฟ?

  • A ๏€น 2 ๐œ‹ โˆ’ 3 ๏… ๐ฟ 6 ๐œ‹ ๏Šจ ๏Šจ ๏Šจ
  • B ๐ฟ 4 ๏Šจ
  • C ๏€น 2 ๐œ‹ โˆ’ 3 ๏… ๐ฟ 1 2 ๐œ‹ ๏Šจ ๏Šฉ ๏Šจ
  • D ๏€น 2 ๐œ‹ โˆ’ 3 ๏… ๐ฟ 1 2 ๐œ‹ ๏Šจ ๏Šจ ๏Šจ
  • E ๏€น ๐œ‹ โˆ’ 4 ๏… ๐ฟ ๐œ‹ ๏Šจ ๏Šฉ ๏Šฉ

Q5:

Which of the following is the expectation value of the kinetic energy for a particle in the state ฮจ(๐‘ฅ,๐‘ก)=๐ด๐‘’๏ƒ(๏‡๏—๏Šฑ๏Šจ๏Ž๏) when confined to a region between 0 and ๐ฟ?

  • A ๐ด ๐‘˜ โ„Ž ๐ฟ 2 ๐‘š ๏Šจ ๏Šจ ๏Šจ
  • B ๐ด ๐‘˜ โ„ ๐ฟ 2 ๐‘š ๏Šจ ๏Šจ ๏Šจ
  • C ๐‘˜ โ„ 2 ๐‘š ๏Šจ ๏Šจ
  • D ๐‘˜ โ„ ๐‘š ๏Šจ ๏Šจ
  • E ๐ด ๐‘˜ โ„ ๐ฟ ๐‘š ๏Šจ ๏Šจ ๏Šจ

Q6:

A free proton has a wave function given by ฮจ(๐‘ฅ,๐‘ก)=๐ด๐‘’๏ƒ๏€บ๏Šช๏Ž–๏Šฆร—๏Šง๏Šฆ๏—๏Šฑ๏Šซ๏Ž–๏Šฆร—๏Šง๏Šฆ๏๏†๏Ž ๏Ž ๏Ž ๏Žค, where ๐‘ฅ is measured in meters and ๐‘ก in seconds.

Find the momentum of the proton.

  • A 7 . 4 ร— 1 0 ๏Šฑ ๏Šง ๏Šฏ kgโ‹…m/s
  • B 4 . 2 ร— 1 0 ๏Šฑ ๏Šจ ๏Šช kgโ‹…m/s
  • C 2 . 7 ร— 1 0 ๏Šฑ ๏Šจ ๏Šฉ kgโ‹…m/s
  • D 4 . 2 ร— 1 0 ๏Šฑ ๏Šจ ๏Šฉ kgโ‹…m/s
  • E 2 . 7 ร— 1 0 ๏Šฑ ๏Šจ ๏Šจ kgโ‹…m/s

Find the energy of the proton.

  • A 8 . 4 ร— 1 0 ๏Šฑ ๏Šจ ๏Šฆ J
  • B 1 . 7 ร— 1 0 ๏Šฑ ๏Šจ ๏Šง J
  • C 5 . 3 ร— 1 0 ๏Šฑ ๏Šง ๏Šฏ J
  • D 5 . 3 ร— 1 0 ๏Šฑ ๏Šจ ๏Šฉ J
  • E 3 . 3 ร— 1 0 ๏Šฑ ๏Šง ๏Šฎ J

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