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๏€ปโˆš9,62,15๏‡โˆ˜โˆ˜
  • B๏€ปโˆš5,62,29๏‡โˆ˜โˆ˜
  • C(6,52,15)โˆ˜โˆ˜
  • D๏€ปโˆš7,69,24๏‡โˆ˜โˆ˜
  • E๏€ปโˆš6,66,27๏‡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›ผ,0,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๏€น2๐œ‹โˆ’3๏…๐ฟ6๐œ‹๏Šจ๏Šจ๏Šจ
  • B๐ฟ4๏Šจ
  • C๏€น2๐œ‹โˆ’3๏…๐ฟ12๐œ‹๏Šจ๏Šฉ๏Šจ
  • D๏€น2๐œ‹โˆ’3๏…๐ฟ12๐œ‹๏Šจ๏Šจ๏Šจ
  • E๏€น๐œ‹โˆ’4๏…๐ฟ๐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?

  • A๐ด๐‘˜โ„Ž๐ฟ2๐‘š๏Šจ๏Šจ๏Šจ
  • B๐ด๐‘˜โ„๐ฟ2๐‘š๏Šจ๏Šจ๏Šจ
  • C๐‘˜โ„2๐‘š๏Šจ๏Šจ
  • D๐‘˜โ„๐‘š๏Šจ๏Šจ
  • E๐ด๐‘˜โ„๐ฟ๐‘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.

  • A7.4ร—10๏Šฑ๏Šง๏Šฏ kgโ‹…m/s
  • B4.2ร—10๏Šฑ๏Šจ๏Šช kgโ‹…m/s
  • C2.7ร—10๏Šฑ๏Šจ๏Šฉ kgโ‹…m/s
  • D4.2ร—10๏Šฑ๏Šจ๏Šฉ kgโ‹…m/s
  • E2.7ร—10๏Šฑ๏Šจ๏Šจ kgโ‹…m/s

Find the energy of the proton.

  • A8.4ร—10๏Šฑ๏Šจ๏Šฆ J
  • B1.7ร—10๏Šฑ๏Šจ๏Šง J
  • C5.3ร—10๏Šฑ๏Šง๏Šฏ J
  • D5.3ร—10๏Šฑ๏Šจ๏Šฉ J
  • E3.3ร—10๏Šฑ๏Šง๏Šฎ J

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